Question ID 82372955

Assessment Test Domain Skill Difficulty

SAT Math Geometry and Circles Hard

Trigonometry

ID: 82372955

In the xy-plane, a circle has center with coordinates . Points and lie on the circle. Point has
coordinates , and is a right angle. What is the length of ?

A.

B.

C.

D.

ID: 82372955 Answer

Correct Answer: A

Rationale

¯ ¯
Choice A is correct. It's given that points 𝐴 and 𝐵 lie on the circle with center 𝐶. Therefore, 𝐴𝐶 and 𝐵𝐶 are both radii of the
¯ ¯ ¯
circle. Since all radii of a circle are congruent, 𝐴𝐶 is congruent to 𝐵𝐶. The length of 𝐴𝐶, or the distance from point 𝐴 to point 𝐶
, can be found using the distance formula, which gives the distance between two points, 𝑥1 , 𝑦1 and 𝑥2 , 𝑦2 , as
𝑥 - 𝑥 2 + 𝑦1 - 𝑦2 2 . Substituting the given coordinates of point 𝐴, ℎ + 1, 𝑘 + √102 , for 𝑥1 , 𝑦1 and the given coordinates of
√ 1 2
2 2
point 𝐶, ℎ, 𝑘, for 𝑥2 , 𝑦2 in the distance formula yields √ℎ + 1 - ℎ2 + 𝑘 + √102 - 𝑘 , or √12 + √102 , which is equivalent to
¯ ¯
√1 + 102 , or √103 . Therefore, the length of 𝐴𝐶 is √103 and the length of 𝐵𝐶 is √103 . It's given that angle 𝐴𝐶𝐵 is a right
¯ ¯ ¯
angle. Therefore, triangle 𝐴𝐶𝐵 is a right triangle with legs 𝐴𝐶 and 𝐵𝐶 and hypotenuse 𝐴𝐵. By the Pythagorean theorem, if a
2
right triangle has a hypotenuse with length 𝑐 and legs with lengths 𝑎 and 𝑏, then 𝑎 + 𝑏 = 𝑐2 . Substituting √103 for 𝑎 and 𝑏
2
2 2
in this equation yields √103 + √103 = 𝑐2 , or 103 + 103 = 𝑐2 , which is equivalent to 206 = 𝑐2 . Taking the positive square
¯
root of both sides of this equation yields √206 = 𝑐. Therefore, the length of 𝐴𝐵 is √206 .

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

¯ ¯
Choice C is incorrect. This would be the length of 𝐴𝐵 if the length of 𝐴𝐶 were 103, not √103 .

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

Question Difficulty: Hard

Question ID 012489f9
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume Hard

Trigonometry

ID: 012489f9

Rectangles and are similar. The length of each side of is times the length of the
corresponding side of . The area of is square units. What is the area, in square units, of
?
A.

B.

C.

D.

ID: 012489f9 Answer

Correct Answer: D

Rationale

Choice D is correct. The area of a rectangle is given by 𝑏ℎ, where 𝑏 is the length of the base of the rectangle and ℎ is its
height. Let 𝑥 represent the length, in units, of the base of rectangle ABCD, and let 𝑦 represent its height, in units. Substituting
𝑥 for 𝑏 and 𝑦 for ℎ in the formula 𝑏ℎ yields 𝑥𝑦. Therefore, the area, in square units, of ABCD can be represented by the
expression 𝑥𝑦. It’s given that the length of each side of EFGH is 6 times the length of the corresponding side of ABCD.
Therefore, the length, in units, of the base of EFGH can be represented by the expression 6𝑥, and its height, in units, can be
represented by the expression 6𝑦. Substituting 6𝑥 for 𝑏 and 6𝑦 for ℎ in the formula 𝑏ℎ yields 6𝑥6𝑦, which is equivalent to
36𝑥𝑦. Therefore, the area, in square units, of EFGH can be represented by the expression 36𝑥𝑦. It’s given that the area of
ABCD is 54 square units. Since 𝑥𝑦 represents the area, in square units, of ABCD, substituting 54 for 𝑥𝑦 in the expression 36𝑥𝑦
yields 3654, or 1,944. Therefore, the area, in square units, of EFGH is 1,944.

1
Choice A is incorrect. This is the area of a rectangle where the length of each side of the rectangle is √ 6 , not 6, times the

length of the corresponding side of ABCD.

2
Choice B is incorrect. This is the area of a rectangle where the length of each side of the rectangle is √ 3 , not 6, times the

length of the corresponding side of ABCD.

Choice C is incorrect. This is the area of a rectangle where the length of each side of the rectangle is √6 , not 6, times the
length of the corresponding side of ABCD.

Question Difficulty: Hard

Question ID f8e6e6c6
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and Hard

Trigonometry trigonometry

ID: f8e6e6c6

An isosceles right triangle has a hypotenuse of length inches. What is the perimeter, in inches, of this
triangle?

A.

B.

C.

D.

ID: f8e6e6c6 Answer

Correct Answer: C

Rationale

Choice C is correct. Since the triangle is an isosceles right triangle, the two sides that form the right angle must be the same
length. Let 𝑥 be the length, in inches, of each of those sides. The Pythagorean theorem states that in a right triangle,
𝑎2 + 𝑏2 = 𝑐2 , where 𝑐 is the length of the hypotenuse and 𝑎 and 𝑏 are the lengths of the other two sides. Substituting 𝑥 for 𝑎,
2 2 582
𝑥 for 𝑏, and 58 for 𝑐 in this equation yields 𝑥2 + 𝑥2 = 58 , or 2𝑥2 = 58 . Dividing each side of this equation by 2 yields 𝑥2 =
2
2 · 582 58√2 58√2
, or 𝑥2 = . Taking the square root of each side of this equation yields two solutions: 𝑥 = and 𝑥 = - . The value
4 2 2
58√2
of 𝑥 must be positive because it represents a side length. Therefore, 𝑥 = , or 𝑥 = 29√2 . The perimeter, in inches, of the
2
triangle is 58 + 𝑥 + 𝑥, or 58 + 2𝑥. Substituting 29√2 for 𝑥 in this expression gives a perimeter, in inches, of 58 + 229√2 , or
58 + 58√2 .

Choice A is incorrect. This is the length, in inches, of each of the congruent sides of the triangle, not the perimeter, in inches,
of the triangle.

Choice B is incorrect. This is the sum of the lengths, in inches, of the congruent sides of the triangle, not the perimeter, in
inches, of the triangle.

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

Question Difficulty: Hard

Question ID aac3872b
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and Easy

Trigonometry triangles

ID: aac3872b

In triangle , the measure of angle is and the measure of angle is . What is the measure of
angle ?
A.

B.

C.

D.

ID: aac3872b Answer

Correct Answer: D

Rationale

Choice D is correct. The sum of the angle measures of a triangle is 180°. Adding the measures of angles 𝐵 and 𝐶 gives
52 + 17 = 69°. Therefore, the measure of angle 𝐴 is 180 - 69 = 111°.

Choice A is incorrect and may result from subtracting the sum of the measures of angles 𝐵 and 𝐶 from 90°, instead of from
180°.

Choice B is incorrect and may result from subtracting the measure of angle 𝐶 from the measure of angle 𝐵.

Choice C is incorrect and may result from adding the measures of angles 𝐵 and 𝐶 but not subtracting the result from 180°.

Question Difficulty: Easy

Question ID 1b0b382b
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and Hard

Trigonometry trigonometry

ID: 1b0b382b

Right triangle is shown. What is the value of ?

A.

B.

C.

D.

ID: 1b0b382b Answer

Correct Answer: C

Rationale

Choice C is correct. In the triangle shown, the measure of angle 𝐵 is 30° and angle 𝐶 is a right angle, which means that it has
a measure of 90°. Since the sum of the angles in a triangle is equal to 180°, the measure of angle 𝐴 is equal to
180° - 30 + 90°, or 60°. In a right triangle whose acute angles have measures 30° and 60°, the lengths of the legs can be
represented by the expressions 𝑥, 𝑥√3 , and 2𝑥, where 𝑥 is the length of the leg opposite the angle with measure 30°, 𝑥√3 is
the length of the leg opposite the angle with measure 60°, and 2𝑥 is the length of the hypotenuse. In the triangle shown, the
hypotenuse has a length of 54. It follows that 2𝑥 = 54, or 𝑥 = 27. Therefore, the length of the leg opposite angle 𝐵 is 27 and
the length of the leg opposite angle 𝐴 is 27√3 . The tangent of an acute angle in a right triangle is defined as the ratio of the
length of the leg opposite the angle to the length of the leg adjacent to the angle. The length of the leg opposite angle 𝐴 is
27√3
27√3 and the length of the leg adjacent to angle 𝐴 is 27. Therefore, the value of tan𝐴 is , or √3 .
27

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

 1
Choice B is incorrect. This is the value of tan𝐴
, not the value of tan𝐴.

Choice D is incorrect. This is the length of the leg opposite angle 𝐴, not the value of tan𝐴.

Question Difficulty: Hard

Question ID 8bca291d
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and Medium

Trigonometry triangles

ID: 8bca291d

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

B.

C.

D.

ID: 8bca291d Answer

Correct Answer: C

Rationale

Choice C is correct. Vertical angles, which are angles that are opposite each other when two lines intersect, are congruent.
The figure shows that lines 𝑡 and 𝑚 intersect. It follows that the angle with measure 𝑥° and the angle with measure 𝑦° are
vertical angles, so 𝑥 = 𝑦. It's given that 𝑥 = 6𝑘 + 13 and 𝑦 = 8𝑘 - 29. Substituting 6𝑘 + 13 for 𝑥 and 8𝑘 - 29 for 𝑦 in the
equation 𝑥 = 𝑦 yields 6𝑘 + 13 = 8𝑘 - 29. Subtracting 6𝑘 from both sides of this equation yields 13 = 2𝑘 - 29. Adding 29 to
both sides of this equation yields 42 = 2𝑘, or 2𝑘 = 42. Dividing both sides of this equation by 2 yields 𝑘 = 21. It's given that
lines 𝑚 and 𝑛 are parallel, and the figure shows that lines 𝑚 and 𝑛 are intersected by a transversal, line 𝑡. If two parallel lines
are intersected by a transversal, then the same-side interior angles are supplementary. It follows that the same-side interior
angles with measures 𝑦° and 𝑧° are supplementary, so 𝑦 + 𝑧 = 180. Substituting 8𝑘 - 29 for 𝑦 in this equation yields
8𝑘 - 29 + 𝑧 = 180. Substituting 21 for 𝑘 in this equation yields 821 - 29 + 𝑧 = 180, or 139 + 𝑧 = 180. Subtracting 139 from
both sides of this equation yields 𝑧 = 41. Therefore, the value of 𝑧 is 41.

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

Choice B is incorrect. This is the value of 𝑘, not 𝑧.

Choice D is incorrect. This is the value of 𝑥 or 𝑦, not 𝑧.

Question Difficulty: Medium

Question ID c3f47bd8
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and Hard

Trigonometry triangles

ID: c3f47bd8

In triangle , angle is a right angle, point lies on , point lies on , and is parallel to .
If the length of is units, the length of is units, and the area of triangle is square
units, what is the length of , in units?

ID: c3f47bd8 Answer

Correct Answer: 14.66, 14.67, 44/3

Rationale

44
The correct answer is 3
. It's given that in triangle 𝑅𝑆𝑇, angle 𝑇 is a right angle. The area of a right triangle can be found using
1
the formula 𝐴 = 𝑙 𝑙 ,
2 1 2
where 𝐴 represents the area of the right triangle, 𝑙1 represents the length of one leg of the triangle, and
¯ ¯
𝑙2 represents the length of the other leg of the triangle. In triangle 𝑅𝑆𝑇, the two legs are 𝑅𝑇 and 𝑆𝑇. Therefore, if the length of
¯ 1
𝑅𝑇 is 72 and the area of triangle 𝑅𝑆𝑇 is 792, then 792 = 72𝑆𝑇, or 792 = 36𝑆𝑇. Dividing both sides of this equation by 36
2
¯ ¯ ¯ ¯
yields 22 = 𝑆𝑇. Therefore, the length of 𝑆𝑇 is 22. It's also given that point 𝐿 lies on 𝑅𝑆, point 𝐾 lies on 𝑆𝑇, and 𝐿𝐾 is parallel to
¯
𝑅𝑇. It follows that angle 𝐿𝐾𝑆 is a right angle. Since triangles 𝑅𝑆𝑇 and 𝐿𝑆𝐾 share angle 𝑆 and have right angles 𝑇 and 𝐾,
¯ ¯
respectively, triangles 𝑅𝑆𝑇 and 𝐿𝑆𝐾 are similar triangles. Therefore, the ratio of the length of 𝑅𝑇 to the length of 𝐿𝐾 is equal to
¯ ¯ ¯ ¯
the ratio of the length of 𝑆𝑇 to the length of 𝑆𝐾. If the length of 𝑅𝑇 is 72 and the length of 𝐿𝐾 is 24, it follows that the ratio of
¯ ¯ 72 ¯ ¯ 22
the length of 𝑅𝑇 to the length of 𝐿𝐾 is 24
, or 3, so the ratio of the length of 𝑆𝑇 to the length of 𝑆𝐾 is 3. Therefore, 𝑆𝐾
= 3.
22
Multiplying both sides of this equation by 𝑆𝐾 yields 22 = 3𝑆𝐾. Dividing both sides of this equation by 3 yields 3
= 𝑆𝐾. Since
¯ ¯ 22 ¯ ¯ 22 44
the length of 𝑆𝑇, 22, is the sum of the length of 𝑆𝐾, 3
, and the length of 𝐾𝑇, it follows that the length of 𝐾𝑇 is 22 - 3
, or 3
.
Note that 44/3, 14.66, and 14.67 are examples of ways to enter a correct answer.

Question Difficulty: Hard

Question ID c5a51dda
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume Hard

Trigonometry

ID: c5a51dda

A cube has a volume of cubic units. What is the surface area, in square units, of the cube?

ID: c5a51dda Answer

Correct Answer: 36504

Rationale

The correct answer is 36,504. The volume of a cube can be found using the formula 𝑉 = 𝑠3 , where 𝑠 represents the edge
length of a cube. It’s given that this cube has a volume of 474,552 cubic units. Substituting 474,552 for 𝑉 in 𝑉 = 𝑠3 yields
474,552 = 𝑠3 . Taking the cube root of both sides of this equation yields 78 = 𝑠. Thus, the edge length of the cube is 78 units.
Since each face of a cube is a square, it follows that each face has an edge length of 78 units. The area of a square can be
2
found using the formula 𝐴 = 𝑠2 . Substituting 78 for 𝑠 in this formula yields 𝐴 = 78 , or 𝐴 = 6,084. Therefore, the area of one
face of this cube is 6,084 square units. Since a cube has 6 faces, the surface area, in square units, of this cube is 66,084, or
36,504.

Question Difficulty: Hard

Question ID a445876d
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and Hard

Trigonometry triangles

ID: a445876d

Triangle is similar to triangle such that , , and correspond to , , and , respectively.

The measure of is and . What is the measure of ?
A.

B.

C.

D.

ID: a445876d Answer

Correct Answer: C

Rationale

Choice C is correct. It’s given that triangle 𝑋𝑌𝑍 is similar to triangle 𝑅𝑆𝑇, such that 𝑋, 𝑌, and 𝑍 correspond to 𝑅, 𝑆, and 𝑇,
respectively. Since corresponding angles of similar triangles are congruent, it follows that the measure of ∠𝑍 is congruent to
the measure of ∠𝑇. It’s given that the measure of ∠𝑍 is 20°. Therefore, the measure of ∠𝑇 is 20°.

Choice A is incorrect and may result from a conceptual error.

Choice B is incorrect. This is half the measure of ∠𝑍.

Choice D is incorrect. This is twice the measure of ∠𝑍.

Question Difficulty: Hard

Question ID ad7bab3b
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and Easy

Trigonometry trigonometry

ID: ad7bab3b

For the right triangle shown, and . Which expression represents the value of ?
A.

B.

C.

D.

ID: ad7bab3b Answer

Correct Answer: D

Rationale

Choice D is correct. By the Pythagorean theorem, if a right triangle has a hypotenuse with length 𝑐 and legs with lengths 𝑎
2
and 𝑏, then 𝑐2 = 𝑎2 + 𝑏 . In the right triangle shown, the hypotenuse has length 𝑐 and the legs have lengths 𝑎 and 𝑏. It's given
2 2
that 𝑎 = 4 and 𝑏 = 5. Substituting 4 for 𝑎 and 5 for 𝑏 in the Pythagorean theorem yields 𝑐2 = 4 + 5 . Taking the square root
2
of both sides of this equation yields 𝑐 = ± √42 + 5 . Since the length of a side of a triangle must be positive, the value of 𝑐
2 2
is √4 + 5 .

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

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

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

Question Difficulty: Easy
Question ID 40789a56
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Circles Hard

Trigonometry

ID: 40789a56

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: 40789a56 Answer

Correct Answer: 16

Rationale

The correct answer is 16. An equation of a circle in the xy-plane can be written as 𝑥 - 𝑡2 + 𝑦 - 𝑢2 = 𝑟2 , where the center of the
circle is 𝑡, 𝑢 , the radius of the circle is 𝑟, and where 𝑡, 𝑢, and 𝑟 are constants. It’s given that the equation of circle A is
2 2 2 2 2
𝑥 + 5 + 𝑦 - 5 = 4, which is equivalent to 𝑥 + 5 + 𝑦 - 5 = 2 . Therefore, the center of circle A is -5, 5 and the radius of circle
A is 2. 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 -5, 5 and the radius of circle B is 22, or 4. Substituting -5 for 𝑡, 5 for 𝑢, and 4 for 𝑟 into the
2 2 2 2 2
equation 𝑥 - 𝑡2 + 𝑦 - 𝑢2 = 𝑟2 yields 𝑥 + 5 + 𝑦 - 5 = 4 , which is equivalent to 𝑥 + 5 + 𝑦 - 5 = 16. It follows that the
2 2
equation of circle B in the xy-plane is 𝑥 + 5 + 𝑦 - 5 = 16. Therefore, the value of 𝑘 is 16.

Question Difficulty: Hard

Question ID f9c5558d
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume Medium

Trigonometry

ID: f9c5558d

Square X has a side length of centimeters. The perimeter of square Y is times the perimeter of square X.
What is the length, in centimeters, of one side of square Y?
A.

B.

C.

D.

ID: f9c5558d Answer

Correct Answer: D

Rationale

Choice D is correct. The perimeter, 𝑃, of a square can be found using the formula 𝑃 = 4𝑠, where 𝑠 is the length of each side of
the square. It's given that square X has a side length of 12 centimeters. Substituting 12 for 𝑠 in the formula for the perimeter
of a square yields 𝑃 = 412, or 𝑃 = 48. Therefore, the perimeter of square X is 48 centimeters. It’s also given that the
perimeter of square Y is 2 times the perimeter of square X. Therefore, the perimeter of square Y is 248, or 96, centimeters.
Substituting 96 for 𝑃 in the formula 𝑃 = 4𝑠 gives 96 = 4𝑠. Dividing both sides of this equation by 4 gives 24 = 𝑠. Therefore,
the length of one side of square Y is 24 centimeters.

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

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

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

Question Difficulty: Medium

Question ID f92d252b
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume Medium

Trigonometry

ID: f92d252b

A right circular cylinder has a volume of cubic centimeters. The area of the base of the cylinder is
square centimeters. What is the height, in centimeters, of the cylinder?

ID: f92d252b Answer

Correct Answer: 29

Rationale

The correct answer is 29. The volume, 𝑉, of a right circular cylinder is given by the formula 𝑉 = 𝜋𝑟2 ℎ, where 𝑟 is the radius of
the base of the cylinder and ℎ is the height of the cylinder. Since the base of the cylinder is a circle with radius 𝑟, the area of
the base of the cylinder is 𝜋𝑟2 . It's given that a right circular cylinder has a volume of 377 cubic centimeters; therefore,
𝑉 = 377. It's also given that the area of the base of the cylinder is 13 square centimeters; therefore, 𝜋𝑟2 = 13. Substituting
377 for 𝑉 and 13 for 𝜋𝑟2 in the formula 𝑉 = 𝜋𝑟2 ℎ yields 377 = 13ℎ. Dividing both sides of this equation by 13 yields 29 = ℎ.
Therefore, the height of the cylinder, in centimeters, is 29.

Question Difficulty: Medium

Question ID 429c2a72
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume Hard

Trigonometry

ID: 429c2a72

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

A.

B.

C.

D.

ID: 429c2a72 Answer

Correct Answer: B

Rationale

1
Choice B is correct. The area, 𝐴, of a triangle can be found using the formula 𝐴 = 2 𝑏ℎ, where 𝑏 is the length of the base of the
triangle and ℎ 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 2√2 , 6√2 , and √80 units. Since √80 units is
the greatest of these lengths, it's the length of the hypotenuse. Therefore, the two legs have lengths 2√2 and 6√2 units.
1 1
Substituting these values for 𝑏 and ℎ in the formula 𝐴 = 𝑏ℎ gives 𝐴 = 2√2 6√2 , which is equivalent to 𝐴 = 6√4 square
2 2
units, or 𝐴 = 12 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 a1060875
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume Easy

Trigonometry

ID: a1060875

A rectangle has a length of and a width of . What is the perimeter of the rectangle?
A.

B.

C.

D.

ID: a1060875 Answer

Correct Answer: C

Rationale

Choice C is correct. The perimeter of a quadrilateral is the sum of the lengths of its four sides. It's given that the rectangle
has a length of 13 and a width of 6. It follows that the rectangle has two sides with length 13 and two sides with length 6.
Therefore, the perimeter of the rectangle is 13 + 13 + 6 + 6, or 38.

Choice A is incorrect. This is the sum of the lengths of the two sides with length 6, not the sum of the lengths of all four
sides of the rectangle.

Choice B is incorrect. This is the sum of the lengths of the two sides with length 13, not the sum of the lengths of all four
sides of the rectangle.

Choice D is incorrect. This is the perimeter of a rectangle that has four sides with length 13, not two sides with length 13 and
two sides with length 6.

Question Difficulty: Easy

Question ID f47594d0
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and Easy

Trigonometry triangles

ID: f47594d0

In the figure shown, line is parallel to line . What is the value of ?

A.

B.

C.

D.

ID: f47594d0 Answer

Correct Answer: D

Rationale

Choice D is correct. The sum of consecutive interior angles between two parallel lines and on the same side of the
transversal is 180 degrees. Since it's given that line 𝑚 is parallel to line 𝑛, it follows that 𝑥 + 26 = 180. Subtracting 26 from
both sides of this equation yields 154. Therefore, the value of 𝑥 is 154.

Choice A is incorrect. This is half of the given angle measure.

Choice B is incorrect. This is the value of the given angle measure.

Choice C is incorrect. This is twice the value of the given angle measure.

Question Difficulty: Easy

Question ID 8aeff54c
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and Hard

Trigonometry trigonometry

ID: 8aeff54c

In the triangle shown, what is the value of ?

ID: 8aeff54c Answer

Correct Answer: .3928, .3929, 11/28

Rationale

11
The correct answer is 28
. The cosine of an acute angle in a right triangle is defined as the ratio of the length of the leg
adjacent to the angle to the length of the hypotenuse. In the triangle shown, the length of the leg adjacent to the angle with
11
measure 𝑥° is 11 units and the length of the hypotenuse is 28 units. Therefore, the value of cos𝑥° is 28
. Note that 11/28,
.3928, .3929, 0.392, and 0.393 are examples of ways to enter a correct answer.

Question Difficulty: Hard

Question ID 51355d23
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and Medium

Trigonometry trigonometry

ID: 51355d23

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: 51355d23 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
𝐹𝐺𝐻 is similar to right triangle 𝐽𝐾𝐿 and angle 𝐹 corresponds to angle 𝐽. It follows that angle 𝐹 is congruent to angle 𝐽 and,
therefore, the measure of angle 𝐹 is equal to the measure of angle 𝐽. The sine ratios of angles of equal measure are equal.
308 308
Since the measure of angle 𝐹 is equal to the measure of angle 𝐽, sin𝐹 = sin𝐽. It's given that sin𝐹 = 317
. Therefore, sin𝐽 is 317
.

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

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

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

Question Difficulty: Medium

Question ID 8e79ef1c
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Circles Medium

Trigonometry

ID: 8e79ef1c

An angle has a measure of radians. What is the measure of the angle in degrees?

ID: 8e79ef1c Answer

Correct Answer: 81

Rationale

The correct answer is 81. The measure of an angle, in degrees, can be found by multiplying its measure, in radians, by
180 degrees 9𝜋 180 degrees 9𝜋 180 degrees
𝜋 radians
. Multiplying the given angle measure, radians, by
20 𝜋 radians
yields 20
radians
𝜋 radians
, which is equivalent to
81 degrees.

Question Difficulty: Medium

Question ID 489aba1c
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume Medium

Trigonometry

ID: 489aba1c

A circle has a circumference of centimeters. What is the diameter, in centimeters, of the circle?

ID: 489aba1c Answer

Correct Answer: 31

Rationale

The correct answer is 31. The circumference of a circle is equal to 2𝜋𝑟 centimeters, where 𝑟 represents the radius, in
centimeters, of the circle, and the diameter of the circle is equal to 2𝑟 centimeters. It's given that a circle has a circumference
of 31𝜋 centimeters. Therefore, 31𝜋 = 2𝜋𝑟. Dividing both sides of this equation by 𝜋 yields 31 = 2𝑟. Since the diameter of the
circle is equal to 2𝑟 centimeters, it follows that the diameter, in centimeters, of the circle is 31.

Question Difficulty: Medium

Question ID 1c55945b
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and Easy

Trigonometry triangles

ID: 1c55945b

In , the measure of is and the measure of is . What is the measure of ?

A.

B.

C.

D.

ID: 1c55945b Answer

Correct Answer: C

Rationale

Choice C is correct. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle
is 180°. It's given that in △ 𝑋𝑌𝑍, the measure of ∠𝑋 is 23° and the measure of ∠𝑌 is 66°. It follows that the measure of ∠𝑍 is
180 - 23 - 66°, or 91°.

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

Choice B is incorrect. This is the sum of the measures of ∠𝑋 and ∠𝑌, not the measure of ∠𝑍.

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

Question Difficulty: Easy

Question ID 8e5cbda2
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and Easy

Trigonometry triangles

ID: 8e5cbda2

In the figure shown, line intersects parallel lines and . What is the value of ?

ID: 8e5cbda2 Answer

Correct Answer: 70

Rationale

The correct answer is 70. Based on the figure, the angle with measure 110° and the angle vertical to the angle with measure
𝑥° are same side interior angles. Since vertical angles are congruent, the angle vertical to the angle with measure 𝑥° also has
measure 𝑥°. It’s given that lines 𝑠 and 𝑡 are parallel. Therefore, same side interior angles between lines 𝑠 and 𝑡 are
supplementary. It follows that 𝑥 + 110 = 180. Subtracting 110 from both sides of this equation yields 𝑥 = 70.

Question Difficulty: Easy

Question ID 50cd2366
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and Hard

Trigonometry trigonometry

ID: 50cd2366

An isosceles right triangle has a perimeter of inches. What is the length, in inches, of one leg of
this triangle?

A.

B.

C.

D.

ID: 50cd2366 Answer

Correct Answer: B

Rationale

Choice B is correct. It's given that the right triangle is isosceles. In an isosceles right triangle, the two legs have equal
lengths, and the length of the hypotenuse is √2 times the length of one of the legs. Let 𝑙 represent the length, in inches, of
each leg of the isosceles right triangle. It follows that the length of the hypotenuse is 𝑙√2 inches. The perimeter of a figure is
the sum of the lengths of the sides of the figure. Therefore, the perimeter of the isosceles right triangle is 𝑙 + 𝑙 + 𝑙√2 inches.
It's given that the perimeter of the triangle is 94 + 94√2 inches. It follows that 𝑙 + 𝑙 + 𝑙√2 = 94 + 94√2 . Factoring the left-
hand side of this equation yields 1 + 1 + √2 𝑙 = 94 + 94√2 , or 2 + √2 𝑙 = 94 + 94√2 . Dividing both sides of this equation by
94 + 94√2
2 + √2 yields 𝑙 = . Rationalizing the denominator of the right-hand side of this equation by multiplying the right-hand
2 + √2
2 - √2 94 + 94√2 2 - √2
side of the equation by 2 - 2 yields 𝑙 = 2 + 2 2 - 2 . Applying the distributive property to the numerator and to the
√ √ √
188 - 94√2 + 188√2 - 94√4 94√2
denominator of the right-hand side of this equation yields 𝑙 = 4 - 2√2 + 2√2 - √4
. This is equivalent to 𝑙 = 2 , or 𝑙 = 47√2
. Therefore, the length, in inches, of one leg of the isosceles right triangle is 47√2 .

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

Choice C is incorrect. This is the length, in inches, of the hypotenuse.

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

Question Difficulty: Hard

Question ID 1dbbea6b
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and Hard

Trigonometry trigonometry

ID: 1dbbea6b

In the triangle shown, what is the value of ?

ID: 1dbbea6b Answer

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

Rationale

16
The correct answer is 23
. 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
16
measure 𝑥° is 16 units and the length of the hypotenuse is 23 units. Therefore, the value of sin𝑥° is 23
. 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 e5cc491b
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and Easy

Trigonometry triangles

ID: e5cc491b

In , the measure of is and the measure of is . What is the measure of ?

A.

B.

C.

D.

ID: e5cc491b Answer

Correct Answer: A

Rationale

Choice A is correct. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle
is 180°. It's given that in △ 𝑋𝑌𝑍, the measure of ∠𝑋 is 24° and the measure of ∠𝑌 is 98°. It follows that the measure of ∠𝑍 is
180 - 24 - 98°, or 58°.

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

Choice C is incorrect. This is the sum of the measures of ∠𝑋 and ∠𝑌, not the measure of ∠𝑍.

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

Question Difficulty: Easy

Question ID 8970ec84
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Right triangles and Hard

Trigonometry trigonometry

ID: 8970ec84

The perimeter of an equilateral triangle is centimeters. The height of this triangle is centimeters,
where is a constant. What is the value of ?

ID: 8970ec84 Answer

Correct Answer: 104

Rationale

The correct answer is 104. An equilateral triangle is a triangle in which all three sides have the same length and all three
angles have a measure of 60°. The height of the triangle, 𝑘√3 , is the length of the altitude from one vertex. The altitude
divides the equilateral triangle into two congruent 30-60-90 right triangles, where the altitude is the side across from the 60°
angle in each 30-60-90 right triangle. Since the altitude has a length of 𝑘√3 , it follows from the properties of 30-60-90 right
triangles that the side across from each 30° angle has a length of 𝑘 and each hypotenuse has a length of 2𝑘. In this case, the
hypotenuse of each 30-60-90 right triangle is a side of the equilateral triangle; therefore, each side length of the equilateral
triangle is 2𝑘. The perimeter of a triangle is the sum of the lengths of each side. It's given that the perimeter of the equilateral
triangle is 624; therefore, 2𝑘 + 2𝑘 + 2𝑘 = 624, or 6𝑘 = 624. Dividing both sides of this equation by 6 yields 𝑘 = 104.

Question Difficulty: Hard

Question ID c655ab2f
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and Easy

Trigonometry triangles

ID: c655ab2f

In the figure, line is parallel to line . What is the value of ?

A.

B.

C.

D.

ID: c655ab2f Answer

Correct Answer: D

Rationale

Choice D is correct. It's given that lines 𝑚 and 𝑛 are parallel. Since line 𝑡 intersects both lines 𝑚 and 𝑛, it's a transversal. The
angles in the figure marked as 134° and 𝑤° are on the same side of the transversal, where one is an interior angle with line 𝑚
as a side, and the other is an exterior angle with line 𝑛 as a side. Thus, the marked angles are corresponding angles. When
two parallel lines are intersected by a transversal, corresponding angles are congruent and, therefore, have equal measure. It
follows that 𝑤° = 134°. Therefore, the value of 𝑤 is 134.

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

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

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

Question Difficulty: Easy
Question ID 919b2d08
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume Medium

Trigonometry

ID: 919b2d08

The figure shows the lengths, in inches, of two sides of a right triangle. What is the area of the triangle, in
square inches?

ID: 919b2d08 Answer

Correct Answer: 7.5, 15/2

Rationale

15 1
The correct answer is 2
. The area, 𝐴, of a triangle is given by the formula 𝐴 = 2 𝑏ℎ, where 𝑏 is the length of the base of the
triangle and ℎ is the height of the triangle. In the right triangle shown, the length of the base of the triangle is 5 inches, and
1 1
the height is 3 inches. It follows that 𝑏 = 5 and ℎ = 3. Substituting 5 for 𝑏 and 3 for ℎ in the formula 𝐴 = 2 𝑏ℎ yields 𝐴 = 2 53,
1 15 15
which is equivalent to 𝐴 = 15, or 𝐴 = . Therefore, the area of the triangle, in square inches, is . Note that 15/2 and 7.5
2 2 2
are examples of ways to enter a correct answer.

Question Difficulty: Medium

Question ID 2384a4cb
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Lines, angles, and Easy

Trigonometry triangles

ID: 2384a4cb

In triangle , and . Which statement is sufficient

to prove that triangle is equilateral?
A.

B.

C.

D.

ID: 2384a4cb Answer

Correct Answer: A

Rationale

Choice A is correct. In an equilateral triangle, all three sides have the same length. It’s given that in triangle 𝐴𝐵𝐶,
𝐴𝐵 = 4,680 mm and 𝐵𝐶 = 4,680 mm. Therefore, if 𝐴𝐶 = 4,680 mm, then all three sides of triangle 𝐴𝐵𝐶 have the same
length, so triangle 𝐴𝐵𝐶 is equilateral. Therefore, 𝐴𝐶 = 4,680 mm is sufficient to prove that triangle 𝐴𝐵𝐶 is equilateral.

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: Easy

Question ID e582b600
Assessment Test Domain Skill Difficulty

SAT Math Geometry and Area and volume Medium

Trigonometry

ID: e582b600

A circle has a radius of meters. What is the area, in square meters, of the circle?
A.

B.

C.

D.

ID: e582b600 Answer

Correct Answer: D

Rationale

Choice D is correct. The area, 𝐴, of a circle is given by the formula 𝐴 = 𝜋𝑟2 , where 𝑟 is the radius of the circle. It’s given that
the circle has a radius of 43 meters. Substituting 43 for 𝑟 in the formula 𝐴 = 𝜋𝑟2 yields 𝐴 = 𝜋432 , or 𝐴 = 1,849𝜋. Therefore,
the area, in square meters, of the circle is 1,849𝜋.

43
Choice A is incorrect. This is the area, in square meters, of a circle with a radius of √ 2 meters.

Choice B is incorrect. This is the area, in square meters, of a circle with a radius of √43 meters.

Choice C is incorrect. This is the circumference, in meters, of the circle.

Question Difficulty: Medium