D5-4

1. 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?

2. 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.

3. 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.

4. 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
5. 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

6. 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.

7. 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?

8. 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

2
9. 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

3
10. The front view and side view of Arrangement A are shown here.

Front View Side View

Below are the front view and side view of some arrangement of unit cubes. Let N denote the largest number
of possible unit cubes in an arrangement with the two views shown below, and let n denote the least number
of possible unit cubes. What is the value of N − n?

Front View Side View

11. If the diameter of a right cylindrical can with circular bases is increased by 25%, by what
percent should the height be increased in order to double the volume of the original can?

12. 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 ?

4
13. 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.

14. 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

15. 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

5
16. 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.

17. 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?

18. 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?

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

20. The diagonals of parallelogram ABCD intersect at E. Point F is the midpoint of segment
BE and H is the midpoint of segment CE. What is the ratio of the area of quadrilateral AF HD to the area
of the parallelogram? Express your answer as a common fraction.

6
21. The lengths indicated on the rectangle shown are in centimeters. What is the number of
square centimeters in the area of the shaded region?

5 9
2
6
6
2
9 5

22. 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”

7
23. 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.

24. 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.

25. A cubical room has edge length 10 feet with A and B denoting two corners that are farthest
apart. A caterpillar crawls from A to B along the walls. In feet, what is the length of the shortest path from
A to B that the caterpillar may have taken? Express your answer in simplest radical form.

26. 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?

27. 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?

8
28. Equilateral triangle ABC has side length 6 cm and is inscribed in circle P. Congruent smaller
circles centered at D, E and F are inscribed in the three regions between an arc of circle P and a side of
△ABC, as shown. If segments AF, BE and CD all intersect P, what is the area of △DEF ? Express your
answer as a common fraction in simplest radical form.

E F

A B
D

29. In each of three boxes are four straws with integer length 1 cm through 4 cm, inclusive. A
straw is randomly selected from each box. What is the probability that a triangle can be formed with the
three straws chosen? Express your answer as a common fraction.

30. The formula for the area of any rectangle with a perimeter of 20 units can be written as
A = m − nd2 , where d is the length of the diagonal and m and n are constants. What is the value of the
product mn?

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