D5-3

1. Triangle ABC has side lengths 9, 10 and 13, with D the midpoint of side BC. What is the
length of segment AD?

2. In the diagram, RECT and LON G are rectangles. How many square units are in the area of
LON G?

T 1O C

3
7
L
N

R E
G

3. A square has sides of length 4 units each. A stripe of width 1 unit is√drawn inside the square,
centered on the diagonal. The area of this stripe can be expressed in the form a + b 2 square units, where a
and b are rational numbers. What is the value of a + b? Express your answer as a decimal to the nearest tenth.

4 1

4. The interior of a right, circular cone is 8 inches tall with a 2-inch radius at the opening. The
interior of the cone is filled with ice cream, and the cone has a hemisphere of ice cream exactly covering the
opening of the cone. What is the volume of ice cream? Express your answer in terms of π.
5. A regular hexagon is inscribed in a circle and another regular hexagon is circumscribed about
the same circle. What is the ratio of the area of the larger hexagon to the area of the smaller hexagon?
Express your answer as a common fraction.

6. An isosceles triangle has two sides of length 7 and an area of 16. What is the product of all
possible values of its perimeter?

7. In square ABCD, AD is 4 centimeters, and M is the midpoint of CD. Let O be the

intersection of AC and BM . What is the ratio of OC to OA? Express your answer as a common fraction.

A B

O
D M C

8. In the figure below, the shaded region is formed by drawing two parallel segments which
connect the midpoints of congruent squares. Each square has side length 1 centimeter. What is the number of
square centimeters in the shaded region? Express your answer as a common fraction.

9. The area of the largest

√ equilateral triangle that can be inscribed in a square of side length 1
unit can be expressed in the form a b − c square units, where a, b, and c are integers. What is the value of
a + b + c?

2
10. A chord of the larger of two concentric circles is tangent to the smaller circle and measures 18
inches. Find the number of square inches in the area of the shaded region. Express your answer in terms of π.

18”

11. In △ABC, ∠C is a right angle. Point M is the midpoint of AB, point N is the midpoint of
AC, and point O is the midpoint of AM . The perimeter of △ABC is 112 cm, and ON = 12.5 cm. What is
the number of square centimeters in the area of quadrilateral M N CB?

12. In right triangle ABC, M and N are midpoints of legs AB and BC, respectively. Leg AB is 6
units long, and leg BC is 8 units long. How many square units are in the area of △AP C?

M
P

B N C

3
13. Four equilateral triangles, △ABG, △BCH, △CDE and △DAF , are constructed inside
square ABCD, as shown. Points E, F, G and H are the vertices of the triangles that lie within square
ABCD. What is the ratio of the area of square EF GH to the area of square ABCD? Express your answer in
simplest radical form.

A B
E

H F

G
D C

14. A circle with center O has radius 8 units and circle P has radius 2 units. The circles are
externally tangent to each other at point Q. Segment T S is the common external tangent to circle O and
circle P at points T and S, respectively. What is the length of segment OS? Express your answer in simplest
radical form.

15. The lengths, in order, of four consecutive sides of an equiangular hexagon are 1, 7, 2 and 4
units, respectively. What is the sum of the lengths of the two remaining sides?

16. A sphere is inscribed in a right

√ cone with base radius 12 cm and height 24 cm, as shown. The
radius of the sphere can be expressed as a c − a cm. What is the value of a + c?

4
17. Three cubes are stacked as shown. If the cubes have edge lengths 1, 2, and 3 as shown, what
is the length of the portion of the segment AB that is contained in the center cube?

A
1

3
B

18. A right triangle has a hypotenuse of 10 m and a perimeter of 22 m. In square meters, what is
the area of the triangle?

19. Each side of an equilateral triangle is 8 inches long. An altitude of this triangle is used as a
side of a square. What is the number of square inches of the area of the square?

20. In circle P with radius 2 units, m∠N P R = 100◦ . If the shaded region has area kπ square
units, what is the value of k? Express your answer as a common fraction.

R
1
N

1 P

5
21. In circle O, shown, OP = 2 units, P L = 8 units, P K = 9 units and N K = 18 units. Points
K, P, and M are collinear, as are points L, P, O, and N. What is the length of segment M N ?

M
O
P

K
L

22. Equilateral △ABC is inscribed in circle O. The radius of circle O is 12 inches. How many
square inches are in the area of △ABC? Express your answer in simplest radical form.

23. The midpoints of the three sides of an equilateral triangle are connected to form a second
triangle. A third triangle is formed by connecting the midpoints of the second triangle. This process is
repeated until a tenth triangle is formed. What is the ratio of the perimeter of the tenth triangle to the
perimeter of the third triangle? Express your answer as a common fraction.

24. A triangle whose side lengths are whole numbers has one side which measures 25 inches and a
perimeter of 80 inches. What is the fewest number of inches that can be the length of one of the remaining
sides?

25. What is the total surface area of the largest regular tetrahedron that can be inscribed inside
of a cube of edge length 1 cm? Express your answer in simplest radical form.

26. In triangle ABC, AB = 12 units and AC = 9 units. Point D is on segment BC so that

BD : DC = 2 : 1. If AD = 6 units, what is the length of segment BC? Express your answer in simplest
radical form.

27. The diagonals of a rectangle intersect at point P . Point P is 5 centimeters further from the
shorter side than the longer side. The perimeter of the rectangle is 44 centimeters. What is the number of
square centimeters in the area of the rectangle?

6
28. The surface area of a rectangular prism is 32 square inches. The volume of the prism is 12
cubic inches. The sum of all edge lengths is 28 inches. If the length, width and height of the prism each are
increased by one inch, what is the volume of the resulting prism, in cubic inches?

29. Kyle will use four identical unit cubes to create a solid. Each cube must be glued to at least
one other cube. Two cubes may only be glued together in such a way that a face of one cube exactly covers a
face of the other cube. How many distinct solids could Kyle create? Two solids are considered to be the same
if one solid can be repositioned to match the other solid.

30. In the figure shown, two lines intersect at a right angle, and two semicircles are drawn so that
each semicircle has its diameter on one line and is tangent to the other line. The larger semicircle has radius
1. The smaller semicircle intersects the larger semicircle, dividing the larger semicircular arc in the ratio 1 : 5.
What is the radius of the smaller semicircle? Express your answer in simplest radical form.

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