D5-5

1. A sphere of radius 4 inches is inscribed in a cone with a base of radius 6 inches. In inches,
what is the height of the cone? Express your answer as a decimal to the nearest tenth.

2. What is the ratio of the area of a square inscribed in a semicircle with radius r to the area of
a square inscribed in a circle with radius r? Express your answer as a common fraction.

3. 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?

4. The area of the largest

5. Triangle ABC has side lengths 9, 10 and 13, with D the midpoint of side BC. What is the
length of segment AD?
6. Triangle ABC is a right isosceles triangle. Points D, E and F are the midpoints of the sides
of the triangle. Point G is the midpoint of segment DF and point H is the midpoint of segment F E. What is
the ratio of the shaded area to the non-shaded area in triangle ABC? Express your answer as a common
fraction.

G F
D

C
B E

7. 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

2
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. 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

3
10. A circle is inscribed in a quarter-circle. The circle has radius r and the quarter-circle has
radius R. Express in simplest radical form r/R.

11. Two sides of scalene △ABC measure 3 centimeters and 5 centimeters. How many different
whole centimeter lengths are possible for the third side?

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

13. 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.

4
14. 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

15. Equilateral triangles are formed by connecting the midpoints of the sides of other equilateral
triangles as shown. How many square inches are in the area of the shaded portion of the figure given that
each side of the largest triangle is 12 inches long? Express your answer as a common fraction in simplest
radical form.

5
16. From a circular piece of paper with radius BC, Jeff removes the unshaded sector shown.
Using the larger shaded sector, he joins edge BC to edge BA (without overlap) to form a cone of radius 12
centimeters and of volume 432π cubic centimeters. What is the number of degrees in the measure of angle
ABC of the sector that is not used?

C
B

17. The three side lengths of a particular triangle are 2, 5, and x units, and the area of the
triangle is x square units. What is the value of x? Express your answer in simplest radical form.

18. What is the ratio of the shaded area to the area of triangle ABC if all the triangles shown are
equilateral?

B C

19. The lengths of the perpendiculars drawn to the sides of a regular hexagon from an interior
point are 4, 5, 6, 8, 9, and 10 centimeters. What is the number of centimeters in the length of a side of this
hexagon? Express your answer as a common fraction in simplest radical form.

6
20. Side AB of regular hexagon ABCDEF is extended past B to point X such that AX = 3AB.
Given that each side of the hexagon is 2 units long, what is the length of segment F X? Express your answer
in simplest radical form.

21. 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.

22. The ratio of the measures of the shortest side of two similar triangles is 2 : 3. The smaller
triangle has an area of 100 square feet. What is the number of square feet in the area of the larger triangle?

23. A medieval weapon in the shape shown consists of 4 arcs which are semicircles of radius 6
inches. Point X is the centroid of the weapon. How many square inches are in the area of the cross-section of
the weapon shown?

24. Circle O has radius 10 units. Point P is on radius OQ and OP = 6 units. How many different
chords containing P , including the diameter, have integer lengths?

O Q
P

7
25. In isosceles trapezoid ABCD, shown here, sides AB and DC are parallel, AB = 10 and
CD = 8. Trapezoids AP QR and BCQP are both similar to trapezoid ABCD. What is the area of trapezoid
ABCD? Express your answer in simplest radical form.

D C
R Q

A P B

26. A diagonal of the front face of a rectangular prism is 13 inches long, and a diagonal of the top
face of the same prism is 15 inches long. The height of the front face of the prism is 5 inches long. How many
cubic inches are in the volume of the prism if each of the dimensions is an integer length?

27. 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”

28. How many times in a 24-hour day will the hour and minute hands of a 12-hour analog clock
form a 90-degree angle?

29. Point P is on AB. Point A has coordinates (3, 11) and point B has coordinates (18, 1). The
ratio of AP : P B = 2 : 3. What is the sum of the coordinates of point P ?

8
30. 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?

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