Question ID e10d8313

Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and

Trigonometry triangles

ID: e10d8313

In the figure shown, points , , , and lie on line segment , and line segment intersects line segment at
point . The measure of is , the measure of is , the measure of is , and the
measure of is . What is the measure, in degrees, of ?

ID: e10d8313 Answer

Correct Answer: 123

Rationale
The correct answer is . The triangle angle sum theorem states that the sum of the measures of the interior angles of
a triangle is degrees. It's given that the measure of is and the measure of is . Since points
, , and form a triangle, it follows from the triangle angle sum theorem that the measure, in degrees, of is
, or . It's also given that the measure of is . Since and are
supplementary angles, the sum of their measures is degrees. It follows that the measure, in degrees, of is
, or . Since points , , and form a triangle, and is the same angle as , it follows from
the triangle angle sum theorem that the measure, in degrees, of is , or . It's given that the
measure of is . Since and are supplementary angles, the sum of their measures is
degrees. It follows that the measure, in degrees, of is , or . Since points , , and form a
triangle, and is the same angle as , it follows from the triangle angle sum theorem that the measure, in
degrees, of is , or .

Question Difficulty: Hard

Question ID bcb66188
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and

Trigonometry trigonometry

ID: bcb66188

Triangle is similar to triangle , where angle corresponds to angle and angles and are right angles.
If , what is the value of ?

A.

B.

C.

D.

ID: bcb66188 Answer

Correct Answer: B

Rationale
Choice B is correct. If two triangles are similar, then their corresponding angles are congruent. It's given that right triangle
is similar to right triangle and angle corresponds to angle . It follows that angle is congruent to angle
and, therefore, the measure of angle is equal to the measure of angle . The sine ratios of angles of equal measure
are equal. Since the measure of angle is equal to the measure of angle , . It's given that
. Therefore, is .

Choice A is incorrect. This is the value of , not the value of .

Choice C is incorrect. This is the reciprocal of the value of , not the value of .

Choice D is incorrect. This is the reciprocal of the value of , not the value of .

Question Difficulty: Medium

Question ID f88f27e5
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and

Trigonometry triangles

ID: f88f27e5
Intersecting lines r, s, and t are shown below.

What is the value of x ?

ID: f88f27e5 Answer

Rationale
The correct answer is 97. The intersecting lines form a triangle, and the angle with measure of is an exterior angle of
this triangle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent
interior angles of the triangle. One of these angles has measure of and the other, which is supplementary to the
angle with measure , has measure of . Therefore, the value of x is .

Question Difficulty: Hard

Question ID e5c57163
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume

Trigonometry

ID: e5c57163

Square A has side lengths that are times the side lengths of square B. The area of square A is times the area of
square B. What is the value of ?

ID: e5c57163 Answer

Correct Answer: 27556

Rationale
The correct answer is . The area of a square is , where is the side length of the square. Let represent the
length of each side of square B. Substituting for in yields . It follows that the area of square B is . It’s given
that square A has side lengths that are times the side lengths of square B. Since represents the length of each
side of square B, the length of each side of square A can be represented by the expression . It follows that the area
of square A is , or . It’s given that the area of square A is times the area of square B. Since the area
of square A is equal to , and the area of square B is equal to , an equation representing the given statement
is . Since represents the length of each side of square B, the value of must be positive. Therefore,
the value of is also positive, so it does not equal . Dividing by on both sides of the equation
yields . Therefore, the value of is .

Question Difficulty: Hard

Question ID 25da87f8
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and

Trigonometry trigonometry

ID: 25da87f8

A triangle with angle measures 30°, 60°, and 90° has a perimeter of .
What is the length of the longest side of the triangle?

ID: 25da87f8 Answer

Rationale
The correct answer is 12. It is given that the triangle has angle measures of 30°, 60°, and 90°, and so the triangle is a
special right triangle. The side measures of this type of special triangle are in the ratio . If x is the measure of the

shortest leg, then the measure of the other leg is and the measure of the hypotenuse is 2x. The perimeter of the

triangle is given to be , and so the equation for the perimeter can be written as .
Combining like terms and factoring out a common factor of x on the left-hand side of the equation gives
. Rewriting the right-hand side of the equation by factoring out 6 gives .

Dividing both sides of the equation by the common factor gives x = 6. The longest side of the right triangle, the
hypotenuse, has a length of 2x, or 2(6), which is 12.

Question Difficulty: Hard

Question ID cf53cb56
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume

Trigonometry

ID: cf53cb56

In the xy-plane shown, square ABCD has its diagonals on the x- and y-axes.
What is the area, in square units, of the square?

A. 20

B. 25

C. 50

D. 100

ID: cf53cb56 Answer

Correct Answer: C

Rationale
Choice C is correct. The two diagonals of square ABCD divide the square into 4 congruent right triangles, where each

triangle has a vertex at the origin of the graph shown. The formula for the area of a triangle is , where b is the
base length of the triangle and h is the height of the triangle. Each of the 4 congruent right triangles has a height of 5

units and a base length of 5 units. Therefore, the area of each triangle is , or 12.5 square units. Since the 4

right triangles are congruent, the area of each is of the area of square ABCD. It follows that the area of the square
ABCD is equal to , or 50 square units.

Choices A and D are incorrect and may result from using 5 or 25, respectively, as the area of one of the 4 congruent right
triangles formed by diagonals of square ABCD. However, the area of these triangles is 12.5. Choice B is incorrect and
may result from using 5 as the length of one side of square ABCD. However, the length of a side of square ABCD is .

Question Difficulty: Medium

Question ID 306264ab
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume

Trigonometry

ID: 306264ab

A right triangle has sides of length , , and units. What is the area of the triangle, in square units?

A.

B.

C.

D.

ID: 306264ab Answer

Correct Answer: B

Rationale
Choice B is correct. The area, , of a triangle can be found using the formula , where is the length of the base
of the triangle and is the height of the triangle. It's given that the triangle is a right triangle. Therefore, its base and
height can be represented by the two legs. It’s also given that the triangle has sides of length , , and units.
Since units is the greatest of these lengths, it's the length of the hypotenuse. Therefore, the two legs have lengths
and units. Substituting these values for and in the formula gives ,
which is equivalent to square units, or square units.

Choice A is incorrect. This expression represents the perimeter, rather than the area, of the triangle.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question Difficulty: Hard

Question ID 6a3fbec3
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and

Trigonometry triangles

ID: 6a3fbec3

In the figure above, and .

What is the length of ?

ID: 6a3fbec3 Answer

Rationale
The correct answer is 4.5. According to the properties of right triangles, BD divides triangle ABC into two similar triangles,
ABD and BCD. The corresponding sides of ABD and BCD are proportional, so the ratio of BD to AD is the same as the ratio

of DC to BD. Expressing this information as a proportion gives . Solving the proportion for DC results in
. Note that 4.5 and 9/2 are examples of ways to enter a correct answer.

Question Difficulty: Hard

Question ID dba6a25a
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and

Trigonometry trigonometry

ID: dba6a25a

In the figure above, is parallel to .

What is the length of ?

ID: dba6a25a Answer

Rationale

The correct answer is 30. In the figure given, since is parallel to and both segments are intersected by , then
angle BDC and angle AEC are corresponding angles and therefore congruent. Angle BCD and angle ACE are also
congruent because they are the same angle. Triangle BCD and triangle ACE are similar because if two angles of one
triangle are congruent to two angles of another triangle, the triangles are similar. Since triangle BCD and triangle ACE are
similar, their corresponding sides are proportional. So in triangle BCD and triangle ACE, corresponds to and

corresponds to . Therefore, . Since triangle BCD is a right triangle, the Pythagorean theorem can be

used to give the value of CD: . Taking the square root of each side gives . Substituting the values

in the proportion yields . Multiplying each side by CE, and then multiplying by yields

. Therefore, the length of is 30.

Question Difficulty: Hard

Question ID f7e626b2
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume

Trigonometry

ID: f7e626b2

The dimensions of a right rectangular prism are 4 inches by 5 inches by

6 inches. What is the surface area, in square inches, of the prism?

A. 30

B. 74

C. 120

D. 148

ID: f7e626b2 Answer

Rationale
Choice D is correct. The surface area is found by summing the area of each face. A right rectangular prism consists of
three pairs of congruent rectangles, so the surface area is found by multiplying the areas of three adjacent rectangles by
2 and adding these products. For this prism, the surface area is equal to , or
, which is equal to 148.

Choice A is incorrect. This is the area of one of the faces of the prism. Choice B is incorrect and may result from adding
the areas of three adjacent rectangles without multiplying by 2. Choice C is incorrect. This is the volume, in cubic inches,
of the prism.

Question Difficulty: Hard

Question ID 694b7fce
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and

Trigonometry triangles

ID: 694b7fce

In the figure shown, units, units, and units. What is the area, in square units, of triangle
?

ID: 694b7fce Answer

Correct Answer: 480

Rationale
The correct answer is . It's given in the figure that angle and angle are right angles. It follows that angle
is congruent to angle . It's also given that angle and angle are the same angle. It follows that
angle is congruent to angle . Since triangles and have two pairs of congruent angles, the
triangles are similar. Sides and in triangle correspond to sides and , respectively, in triangle
. Corresponding sides in similar triangles are proportional. Therefore, . It's given that units
and units. Therefore, units. It’s also given that units. Substituting for , for
, and for in the equation yields , or . Multiplying each side of this equation by
yields . By the Pythagorean theorem, if a right triangle has a hypotenuse with length and legs with
lengths and , then . Since triangle is a right triangle, it follows that represents the length of
the hypotenuse, , and and represent the lengths of the legs, and . Substituting for and for in

the equation yields , which is equivalent to , or

. Subtracting from both sides of this equation yields . Taking the square root of both
sides of this equation yields . Since represents a length, which must be positive, the value of is .
Therefore, . Since and represent the lengths of the legs of triangle , it follows that and
can be used to calculate the area, in square units, of the triangle as , or . Therefore, the area, in square
units, of triangle is .
Question Difficulty: Hard
Question ID 17912810
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and

Trigonometry triangles

ID: 17912810

In the figure, parallel lines and are intersected by lines and . If and , what is the value of ?

ID: 17912810 Answer

Correct Answer: 101/2, 50.5

Rationale
The correct answer is . In the figure, lines , , and form a triangle. One interior angle of this triangle is vertical to
the angle marked ; therefore, the interior angle also has measure . It's given that . Therefore, the interior
angle of the triangle has measure . A second interior angle of the triangle forms a straight line, , with the angle
marked . Therefore, the sum of the measures of these two angles is . It's given that . Therefore, the angle
marked has measure and the second interior angle of the triangle has measure , or . The sum
of the interior angles of a triangle is . Therefore, the measure of the third interior angle of the triangle is
, or . It's given that parallel lines and are intersected by line . It follows that the triangle's
interior angle with measure is congruent to the same side interior angle between lines and formed by lines and
. Since this angle is supplementary to the two angles marked , the sum of , , and is . It follows that
, or . Subtracting from both sides of this equation yields . Dividing
both sides of this equation by yields . Note that 101/2 and 50.5 are examples of ways to enter a correct
answer.

Question Difficulty: Hard

Question ID 13d9a1c3
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and

Trigonometry trigonometry

ID: 13d9a1c3

In the right triangle shown above, what

is the length of ?

ID: 13d9a1c3 Answer

Rationale
The correct answer is 4. Triangle PQR has given angle measures of 30° and 90°, so the third angle must be 60° because
the measures of the angles of a triangle sum to 180°. For any special right triangle with angles measuring 30°, 60°, and
90°, the length of the hypotenuse (the side opposite the right angle) is 2x, where x is the length of the side opposite the
30° angle. Segment PQ is opposite the 30° angle. Therefore, 2(PQ) = 8 and PQ = 4.

Question Difficulty: Medium

Question ID 099526fc
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume

Trigonometry

ID: 099526fc

The line segment shown in the xy-plane represents one of the legs of a right triangle. The area of this triangle is
square units. What is the length, in units, of the other leg of this triangle?

A.

B.

C.

D.

ID: 099526fc Answer

Correct Answer: B
Rationale
Choice B is correct. The length of a segment in the xy-plane can be found using the distance formula,

, where and are the endpoints of the segment. The segment shown has
endpoints at and . Substituting and for and , respectively, in the

distance formula yields , or , which is equivalent to , or . Let

represent the length, in units, of the other leg of this triangle. The area, , of a right triangle can be calculated using the
formula , where and are the lengths of the legs of the triangle. It's given that the area of the triangle is
square units. Substituting for , for , and for in the formula yields
. Multiplying both sides of this equation by yields . Dividing both sides of

this equation by yields . Multiplying the numerator and denominator on the left-hand side of this
equation by yields , or , which is equivalent to , or . Therefore, the
length, in units, of the other leg of this triangle is .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. is equivalent to , which is the length, in units, of the line segment shown in the xy-
plane, not the length, in units, of the other leg of the triangle.

Choice D is incorrect and may result from conceptual or calculation errors.

Question Difficulty: Hard

Question ID ae041e52
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and

Trigonometry trigonometry

ID: ae041e52

A square is inscribed in a circle. The radius of the circle is inches. What is the side length, in inches, of the square?

A.

B.

C.

D.

ID: ae041e52 Answer

Correct Answer: A

Rationale
Choice A is correct. When a square is inscribed in a circle, a diagonal of the square is a diameter of the circle. It's given
that a square is inscribed in a circle and the length of a radius of the circle is inches. Therefore, the length of a
diameter of the circle is inches, or inches. It follows that the length of a diagonal of the square is
inches. A diagonal of a square separates the square into two right triangles in which the legs are the sides of the square
and the hypotenuse is a diagonal. Since a square has congruent sides, each of these two right triangles has congruent
legs and a hypotenuse of length inches. Since each of these two right triangles has congruent legs, they are both
- - triangles. In a - - triangle, the length of the hypotenuse is times the length of a leg. Let represent
the length of a leg of one of these - - triangles. It follows that . Dividing both sides of this
equation by yields . Therefore, the length of a leg of one of these - - triangles is inches. Since the
legs of these two - - triangles are the sides of the square, it follows that the side length of the square is inches.

Choice B is incorrect. This is the length of a radius, in inches, of the circle.

Choice C is incorrect. This is the length of a diameter, in inches, of the circle.

Choice D is incorrect and may result from conceptual or calculation errors.

Question Difficulty: Hard

Question ID fb58c0db
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Circles

Trigonometry

ID: fb58c0db

Points A and B lie on a circle with radius 1, and arc has length . What

fraction of the circumference of the circle is the length of arc ?

ID: fb58c0db Answer

Rationale

The correct answer is . The circumference, C, of a circle is , where r is the length of the radius of the circle.

For the given circle with a radius of 1, the circumference is , or . To find what fraction of the

circumference the length of arc is, divide the length of the arc by the circumference, which gives . This

division can be represented by . Note that 1/6, .1666, .1667, 0.166, and 0.167 are examples of ways to
enter a correct answer.

Question Difficulty: Hard

Question ID f243c383
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume

Trigonometry

ID: f243c383

Two identical rectangular prisms each have a height of . The base of each prism is a square, and
the surface area of each prism is . If the prisms are glued together along a square base, the resulting prism has a
surface area of . What is the side length, in , of each square base?

A.

B.

C.

D.

ID: f243c383 Answer

Correct Answer: B

Rationale
Choice B is correct. Let represent the side length, in , of each square base. If the two prisms are glued together
along a square base, the resulting prism has a surface area equal to twice the surface area of one of the prisms, minus
the area of the two square bases that are being glued together, which yields . It’s given that this resulting
surface area is equal to , so . Subtracting from both sides of this equation yields
. This equation can be rewritten by multiplying on the left-hand side by , which yields
, or . Adding to both sides of this equation yields .
Multiplying both sides of this equation by yields . The surface area , in , of each rectangular prism
is equivalent to the sum of the areas of the two square bases and the areas of the four lateral faces. Since the height of
each rectangular prism is and the side length of each square base is , it follows that the area of each square
base is and the area of each lateral face is . Therefore, the surface area of each rectangular prism can
be represented by the expression , or . Substituting this expression for in the equation
yields . Subtracting and from both sides of this equation yields
. Factoring from the right-hand side of this equation yields . Applying the zero
product property, it follows that and . Adding to both sides of the equation
yields . Dividing both sides of this equation by yields . Since a side length of a rectangular prism
can’t be , the length of each square base is .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question Difficulty: Hard

Question ID 7c25b0dc
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and

Trigonometry trigonometry

ID: 7c25b0dc

The length of a rectangle’s diagonal is , and the length of the rectangle’s shorter side is . What is the length of the
rectangle’s longer side?

ID: 7c25b0dc Answer

Correct Answer: 12

Rationale
The correct answer is . The diagonal of a rectangle forms a right triangle, where the shorter side and the longer side of
the rectangle are the legs of the triangle and the diagonal of the rectangle is the hypotenuse of the triangle. It's given that
the length of the rectangle's diagonal is and the length of the rectangle's shorter side is . Thus, the length of the
hypotenuse of the right triangle formed by the diagonal is and the length of one of the legs is . By the
Pythagorean theorem, if a right triangle has a hypotenuse with length and legs with lengths and , then

. Substituting for and for in this equation yields , or . Subtracting

from both sides of this equation yields . Taking the square root of both sides of this equation yields
, or . Since represents a length, which must be positive, the value of is . Thus, the length of
the rectangle's longer side is .

Question Difficulty: Hard

Question ID e80d62c6
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Circles

Trigonometry

ID: e80d62c6

The equation represents circle A. Circle B is obtained by shifting circle A down units in the xy-
plane. Which of the following equations represents circle B?

A. msup

B. msup

C. msup

D. msup

ID: e80d62c6 Answer

Correct Answer: A

Rationale
Choice A is correct. The standard form of an equation of a circle in the xy-plane is , where the
coordinates of the center of the circle are and the length of the radius of the circle is . The equation of circle A,
, can be rewritten as . Therefore, the center of circle A is at and
the length of the radius of circle A is . If circle A is shifted down units, the y-coordinate of its center will decrease by ;
the radius of the circle and the x-coordinate of its center will not change. Therefore, the center of circle B is at ,
or , and its radius is . Substituting for , for , and for in the equation
yields , or . Therefore, the equation
represents circle B.

Choice B is incorrect. This equation represents a circle obtained by shifting circle A up, rather than down, units.

Choice C is incorrect. This equation represents a circle obtained by shifting circle A right, rather than down, units.

Choice D is incorrect. This equation represents a circle obtained by shifting circle A left, rather than down, units.

Question Difficulty: Hard

Question ID a07ed090
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume

Trigonometry

ID: a07ed090

The figure shown is a right circular cylinder with a radius of and height of . A second right circular cylinder (not
shown) has a volume that is times as large as the volume of the cylinder shown. Which of the following could
represent the radius , in terms of , and the height , in terms of , of the second cylinder?

A. and

B. and

C. and

D. and

ID: a07ed090 Answer

Correct Answer: C
Rationale
Choice C is correct. The volume of a right circular cylinder is equal to , where is the radius of a base of the cylinder
and is the height of the cylinder. It’s given that the cylinder shown has a radius of and a height of . It follows that the
volume of the cylinder shown is equal to . It’s given that the second right circular cylinder has a radius of and a
height of . It follows that the volume of the second cylinder is equal to . Choice C gives and .
Substituting for and for in the expression that represents the volume of the second cylinder yields
, or , which is equivalent to , or . This expression is equal to times
the volume of the cylinder shown, . Therefore, and could represent the radius , in terms of , and
the height , in terms of , of the second cylinder.

Choice A is incorrect. Substituting for and for in the expression that represents the volume of the second
cylinder yields , or , which is equivalent to , or . This expression is equal
to , not , times the volume of the cylinder shown.

Choice B is incorrect. Substituting for and for in the expression that represents the volume of the second
cylinder yields , or , which is equivalent to , or . This expression
is equal to , not , times the volume of the cylinder shown.

Choice D is incorrect. Substituting for and for in the expression that represents the volume of the second
cylinder yields , or , which is equivalent to , or . This
expression is equal to , not , times the volume of the cylinder shown.

Question Difficulty: Hard

Question ID 858fd1cf
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Circles

Trigonometry

ID: 858fd1cf

A circle in the xy-plane has its center at . Line is tangent to this circle at the point . Which of the
following points also lies on line ?

A.

B.

C.

D.

ID: 858fd1cf Answer

Correct Answer: C

Rationale
Choice C is correct. It’s given that the circle has its center at and that line is tangent to this circle at the point
. Therefore, the points and are the endpoints of the radius of the circle at the point of tangency.
The slope of a line or line segment that contains the points and can be calculated as . Substituting
for and for in the expression yields , or . Thus, the slope of this radius is
. A line that’s tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It follows that line
is perpendicular to the radius at the point , so the slope of line is the negative reciprocal of the slope of this
radius. The negative reciprocal of is . Therefore, the slope of line is . Since the slope of line is the same
between any two points on line , a point lies on line if the slope of the line segment connecting the point and is
. Substituting choice C, , for and for in the expression yields , or . Therefore,
the point lies on line .

Choice A is incorrect. The slope of the line segment connecting and is , or , not .

Choice B is incorrect. The slope of the line segment connecting and is , or , not .

Choice D is incorrect. The slope of the line segment connecting and is , or , not .

Question Difficulty: Hard

Question ID 9966235e
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume

Trigonometry

ID: 9966235e

A cube has an edge length of inches. A solid sphere with a radius of inches is inside the cube, such that the sphere
touches the center of each face of the cube. To the nearest cubic inch, what is the volume of the space in the cube not
taken up by the sphere?

A.

B.

C.

D.

ID: 9966235e Answer

Correct Answer: A

Rationale
Choice A is correct. The volume of a cube can be found by using the formula , where is the volume and is the
edge length of the cube. Therefore, the volume of the given cube is , or cubic inches. The volume of a
sphere can be found by using the formula , where is the volume and is the radius of the sphere.
Therefore, the volume of the given sphere is , or approximately cubic inches. The volume of the
space in the cube not taken up by the sphere is the difference between the volume of the cube and volume of the sphere.
Subtracting the approximate volume of the sphere from the volume of the cube gives
cubic inches.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question Difficulty: Hard

Question ID 901c3215
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and

Trigonometry triangles

ID: 901c3215

In triangles and , angles and each have measure and angles and each have measure .
Which additional piece of information is sufficient to determine whether triangle is congruent to triangle ?

A. The measure of angle

B. The length of side

C. The lengths of sides and

D. No additional information is necessary.

ID: 901c3215 Answer

Correct Answer: C

Rationale
Choice C is correct. Since angles and each have the same measure and angles and each have the same
measure, triangles and are similar, where side corresponds to side . To determine whether two
similar triangles are congruent, it is sufficient to determine whether one pair of corresponding sides are congruent.
Therefore, to determine whether triangles and are congruent, it is sufficient to determine whether sides
and have equal length. Thus, the lengths of and are sufficient to determine whether triangle is
congruent to triangle .

Choice A is incorrect and may result from conceptual errors. Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect. The given information is sufficient to determine that triangles and are similar, but not
whether they are congruent.

Question Difficulty: Hard

Question ID 2d2cb85e
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and

Trigonometry triangles

ID: 2d2cb85e

In the figure, , the measure of angle is , and the measure of angle is . What is the value of
?

ID: 2d2cb85e Answer

Correct Answer: 156

Rationale
The correct answer is . In the figure shown, the sum of the measures of angle and angle is . It’s
given that the measure of angle is . Therefore, the measure of angle is , or . The sum
of the measures of the interior angles of a triangle is . In triangle , the measure of angle is and it's
given that the measure of angle is . Thus, the measure of angle is , or . It’s given
that . Therefore, triangle is an isosceles triangle and the measure of is equal to the measure of
angle . In triangle , the measure of angle is and the measure of angle is . Thus, the
measure of angle is , or . The figure shows that the measure of angle is , so the
value of is .

Question Difficulty: Hard

Question ID c8345903
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Circles

Trigonometry

ID: c8345903

The circle above has center O, the length of arc is , and

. What is the length of arc ?

A.

B.

C.

D.

ID: c8345903 Answer

Correct Answer: B

Rationale
Choice B is correct. The ratio of the lengths of two arcs of a circle is equal to the ratio of the measures of the central
angles that subtend the arcs. It’s given that arc is subtended by a central angle with measure 100°. Since the sum

of the measures of the angles about a point is 360°, it follows that arc is subtended by a central angle with

measure . If s is the length of arc , then s must satisfy the ratio . Reducing the

fraction to its simplest form gives . Therefore, . Multiplying both sides of by

yields .

Choice A is incorrect. This is the length of an arc consisting of exactly half of the circle, but arc is greater than half
of the circle. Choice C is incorrect. This is the total circumference of the circle. Choice D is incorrect. This is half the
length of arc , not its full length.

Question Difficulty: Hard

Question ID 345cc36a
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and

Trigonometry triangles

ID: 345cc36a

In the figure shown, and intersect at point . , , , and . What is the

length of ?

ID: 345cc36a Answer

Correct Answer: 54

Rationale
The correct answer is . The figure shown includes two triangles, triangle and triangle , such that angle
and angle are vertical angles. It follows that angle is congruent to angle . It’s also given in
the figure that the measures of angle and angle are . Therefore angle is congruent to angle . Since triangle
and triangle have two pairs of congruent angles, triangle is similar to triangle by the angle-
angle similarity postulate, where corresponds to , and corresponds to . Since the lengths of
corresponding sides in similar triangles are proportional, it follows that . It’s given that ,
, and . Substituting for , for , and for in the equation yields
. Multiplying each side of this equation by yields , or . Therefore, the length of
is .

Question Difficulty: Hard

Question ID e50afdd3
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Circles

Trigonometry

ID: e50afdd3

The graph of the given equation is a circle in the xy-plane. The point lies on the circle. Which of the following is a
possible value for ?

A.

B.

C.

D.

ID: e50afdd3 Answer

Correct Answer: B

Rationale
Choice B is correct. An equation of the form , where , , and are constants, represents a
circle in the xy-plane with center and radius . Therefore, the circle represented by the given equation has center
and radius . Since the center of the circle has an x-coordinate of and the radius of the circle is , the
least possible x-coordinate for any point on the circle is , or . Similarly, the greatest possible x-coordinate
for any point on the circle is , or . Therefore, if the point lies on the circle, it must be true that
. Of the given choices, only satisfies this inequality.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question Difficulty: Hard

Question ID 76c73dbf
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Circles

Trigonometry

ID: 76c73dbf

The graph of in the xy-plane is a circle. What is the length of the circle’s radius?

ID: 76c73dbf Answer

Correct Answer: 10

Rationale
The correct answer is . It's given that the graph of in the xy-plane is a circle. The equation of
a circle in the xy-plane can be written in the form , where the coordinates of the center of the
circle are and the length of the radius of the circle is . The term in this equation can be obtained by
adding the square of half the coefficient of to both sides of the given equation to complete the square. The coefficient
of is . Half the coefficient of is . The square of half the coefficient of is . Adding to each side of
yields , or .
Similarly, the term can be obtained by adding the square of half the coefficient of to both sides of this
equation, which yields , or .
This equation is equivalent to , or . Therefore, the length
of the circle's radius is .

Question Difficulty: Hard

Question ID b8a225ff
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Circles

Trigonometry

ID: b8a225ff

Circle A in the xy-plane has the equation . Circle B has the same center as circle A. The radius
of circle B is two times the radius of circle A. The equation defining circle B in the xy-plane is ,
where is a constant. What is the value of ?

ID: b8a225ff Answer

Correct Answer: 16

Rationale
The correct answer is . An equation of a circle in the xy-plane can be written as , where the
center of the circle is , the radius of the circle is , and where , , and are constants. It’s given that the equation
of circle A is , which is equivalent to . Therefore, the center of
circle A is and the radius of circle A is . It’s given that circle B has the same center as circle A and that the
radius of circle B is two times the radius of circle A. Therefore, the center of circle B is and the radius of circle B
is , or . Substituting for , for , and for into the equation yields
, which is equivalent to . It follows that the equation of circle B in
the xy-plane is . Therefore, the value of is .

Question Difficulty: Hard

Question ID ebbf23ae
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Circles

Trigonometry

ID: ebbf23ae

A circle in the xy-plane has a diameter with endpoints and . An equation of this circle is
, where is a positive constant. What is the value of ?

ID: ebbf23ae Answer

Correct Answer: 5

Rationale
The correct answer is . The standard form of an equation of a circle in the xy-plane is ,
where , , and are constants, the coordinates of the center of the circle are , and the length of the radius of the
circle is . It′s given that an equation of the circle is . Therefore, the center of this circle is
. It’s given that the endpoints of a diameter of the circle are and . The length of the radius is the
distance from the center of the circle to an endpoint of a diameter of the circle, which can be found using the distance

formula, . Substituting the center of the circle and one endpoint of the diameter

in this formula gives a distance of , or , which is equivalent to . Since the

distance from the center of the circle to an endpoint of a diameter is , the value of is .

Question Difficulty: Hard

Question ID 1429dcdf
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and

Trigonometry trigonometry

ID: 1429dcdf

In the triangle shown, what is the value of ?

ID: 1429dcdf Answer

Correct Answer: .6956, .6957, 16/23

Rationale
The correct answer is . In a right triangle, the sine of an acute angle is defined as the ratio of the length of the side
opposite the angle to the length of the hypotenuse. In the triangle shown, the length of the side opposite the angle with
measure is units and the length of the hypotenuse is units. Therefore, the value of is . Note that
16/23, .6956, .6957, 0.695, and 0.696 are examples of ways to enter a correct answer.

Question Difficulty: Hard

Question ID b1e1c2f5
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and

Trigonometry triangles

ID: b1e1c2f5

In right triangle , angle is the right angle and . Point on side is connected by a line segment
with point on side such that line segment is parallel to side and . What is the length of line
segment ?

ID: b1e1c2f5 Answer

Correct Answer: 54

Rationale
The correct answer is . It’s given that in triangle , point on side is connected by a line segment with point
on side such that line segment is parallel to side . It follows that parallel segments and are
intersected by sides and . If two parallel segments are intersected by a third segment, corresponding angles are
congruent. Thus, corresponding angles and are congruent and corresponding angles and are
congruent. Since triangle has two angles that are each congruent to an angle in triangle , triangle is
similar to triangle by the angle-angle similarity postulate, where side corresponds to side , and side
corresponds to side . Since the lengths of corresponding sides in similar triangles are proportional, it follows that
. Since point lies on side , . It's given that . Substituting for
in the equation yields , or . It’s given that . Substituting
for and for in the equation yields , or . Multiplying both sides of
this equation by yields . Thus, the length of line segment is .

Question Difficulty: Hard