Download March 24, 2011

March 24, 2011
2011 State Math Contest
Wake Technical Community College
Geometry Test
1. When a square is cut into two congruent rectangles, each has a perimeter of P feet. When the square is
cut into three congruent rectangles, each has a perimeter of P − 6 feet. Determine the area of the square.
a. 144 ft2
b. 324 ft2
c. 576 ft2
d. 900 ft2
e. 1296 ft2
2. A circle is inscribed in a square of diagonal length 12 inches. What is the area of the circle?
a. 18 sq in
b. 72π sq in
c. 36π sq in
3. Let
be nine times
and the measure of the supplement of
the supplement of
. Determine the
.
a. 8°
b. 10°
c. 12°
d. 18π sq in
e. 36 sq in
be nine times the measure of
d. 15°
e. 18°
4. A unit square is translated 4 units to the right and then 4 units down and then directly back to the
starting position. What is the total area swept out by the traveling square? Some locations are “swept
more than once; only count them once in the total area.
a. 16 units2
b. 20 units2
c. 15 units2
d. 10 units2
e. 18 units2
d. 153
e. 85
5. Compute the maximum number of intersections of 17 straight lines.
a. 136
b. 120
c. 128
6. Compute the positive difference between the sum of the interior angles and the sum of the exterior
angles of a regular octagon.
a. 1080°
b. 900°
c. 720°
d. 810°
e. 990°
7. The top of the Leaning Tower of Pisa is 55.86 meters vertically above the ground. The top of the tower
is currently displaced 3.9 meters horizontally. Based on these measurements estimate the length of the
tower to the nearest hundredth of a meter.
a. 56.00 m
b. 56.08 m
c. 55.98 m
1
d. 55.99 m
e. 56.8 m
2011 State Math Contest
Wake Technical Community College
Geometry Test
8. Let F be the height of an equilateral triangle and let K be the area of the triangle. Determine
√3
2
a. 2F2
b. 3.5F2
c. 1.5F2
d. 0.25F2
e. 0.5F2
9. Two similar triangles have areas of 80 in2 and 64 in2. If one side of the smaller triangle is 10 inches,
what is the length of the corresponding side of the larger triangle to the nearest tenth of an inch?
a. Not enough information
b. 11.2 in
c. 12.5 inch
d. 11.8 in
e. 12.1 in
10. In triangle ABC the angle bisector of angle A intersects side BC in the point D. Let BC = 12 cm,
BD = 8 cm, and AC = 3 cm. What is the length of side AB?
a. 6 cm
b. 3.5 cm
c. 4 cm
d. 4.5 cm
e. 1.5 cm
11. Let P and Q be two concentric circles of radii 8 in and 3 in, respectively. Determine the area of the
region inside circle P and outside of a square inscribed in Q to the nearest square inch.
a. 165 in2
b. 178 in2
c. 182 in2
d. 155 in2
e. 183 in2
12. A solid cube has edges of length one foot. The set of points precisely one foot away from the surface of
the cube encloses a solid shape. Determine the volume of this shape to the nearest cubic foot.
a. 7 ft3
b. 21 ft3
c. 15 ft3
d. 20 ft3
e. 17 ft3
13. A trapezoidal water trough is 10 feet long and 2 feet deep, see diagram. The lengths of each base of the
isosceles trapezoid are 2 feet and 6 feet, respectively. What is the
water depth when the trough is holding 12.5 cubic feet of water?
a. 1.5 ft
b. 0.75 ft
c. 0.25 ft
2
d. 0.625 ft
e. 0.5 ft
2011 State Math Contest
Wake Technical Community College
Geometry Test
14. Eighteen segments of length 2.5 inches are joined at their endpoints to create a rectangle. What is the
largest possible area of the rectangle?
a. 50 in2
b. 46 in2
c. 156.25 in2
d. 125 in2
e. 100 in2
15. A square floor is covered with square tiles. If the total number of tiles in the two diagonals is 37, how
many tiles cover the floor?
a. 361
b. 324
c. 289
16. If a square has side of length S and a diagonal of length D, what is
a. 2√2
b. 1
c. 1.5
d. 400
e. 342
?
d. 2
e. 4
17. The measures of the angles of a triangle are represented by 3x, 2y, and 3y – 3x. Which of the following
can be the sum of x and y if the triangle is isosceles?
a. 48°
b. 72°
c. 40°
d. 50°
e. 108°
18. Let P and Q be externally tangent circles of radii 9 inches and 3 inches and let AB be tangent to both P
and Q at points A and B, respectively. What is the area of
quadrilateral APQB?
a. 72 in2
b. 36√3 in2
c. 36 in2
3
d. 72√2 in2
e. 72√3 in2
2011 State Math Contest
Wake Technical Community College
Geometry Test
19. Cube A has a surface area that is 125% of the surface area of cube B. If the volume of cube B is x% of
the volume of cube A, determine x to the nearest integer.
a. 68
b. 70
c. 72
d. 75
e. 76
20. Determine the degree measure to the nearest minute of the central angle that has an intercepted arc
measuring 15 ft in a circle of diameter 19 ft.
a. 89°54´
b. 91°34´
c. 90°12´
d. 90°28´
e. 90°47´
21. A target is made up of three concentric circles whose radii are in the ratio 1:2:3. What is the probability
that a random shot that hits the target will hit inside the second circle but outside the innermost circle?
a.
b.
c.
d.
e.
22. In right triangle RST, angle R is a right angle and RS = 2RT. Point D is chosen on side ST, and
perpendicular lines are drawn to sides RS and RT, intersecting at points E and F, respectively. If REDF
is a square, determine the ratio of SD to DT.
a.
b. 4
c.
d. √2
√
e. 2
23. A point A = (1, 4) is reflected about the line y = x to the point B. Next the point B is reflected about the
line y = − x to a point C. What is the area of triangle ABC?
a. 14.5 units2
b. 16 units2
c. 15 units2
d. 20 units2
e. 14 units2
24. In triangle ABC side AB is the same length as side AC. The angle bisector of angle B intersects side AC
in the point D and the angle bisector of angle BDA intersects side AB in the point E. Determine the ratio
of
a.
b.
d. 1.5
c.
4
e. 3
2011 State Math Contest
Wake Technical Community College
Geometry Test
25. How many different 3-letter strings can be formed from the letters of GEOMETRY (no letter can be
used in a given string more times than it appears in the word)?
a. 336
b. 228
c. 218
5
d. 168
e. 210
2011 State Math Contest
Wake Technical Community College
SHORT ANSWER
Place the answer in the appropriate space.
Geometry Test
66. A triangle with sides 26 cm, 28 cm, and 30 cm is constructed so that the longest and shortest sides are
tangent to a circle. The third side passes through the center of the circle. Determine the radius of the
circle.
67. A rectangular prism has a height of 4 in. Determine the length and width of the prism if all the edges
are integers, the numerical values of the surface area and volume are the same, and the volume is
minimized.
68. The center of a sphere of radius 4 feet is a distance of 2 feet from a plane intersecting the sphere. What
is the area of the circle of intersection of the plane and the sphere to the nearest tenth of a square foot?
69. Side BC of an equilateral triangle ABC is extended through the point B to a point D such that the
distance from D to side AC is 10 feet and the distance from D to side AB extended is 4 feet. What is the
height of triangle ABC to the nearest foot?
70. Let ABGFHCDE be the vertices of the cube pictured below with sides of length 3 yards and let K be
the midpoint of edge HE. Determine the length of BK to the nearest tenth of a yard.
6
2011 State Math Contest
Wake Technical Community College
Geometry Test
1. b
2. d
3. e
4. c
5. a
6. c
7. a
8. e
9. b
10. a
11. e
12. b
13. e
14. d
15. a
16. d
17. a
18. b
19. c
20. d
21. e
22. e
23. c
24. c
25. d
66. 12 cm
67. 8 in by 8 in
68. 37.7 ft2
69. 6 ft
70. 4.5 yd
7
2011 State Math Contest
Wake Technical Community College
Geometry Test
1. Let 3x be the length of the side of the square. Then the conditions give 3
3
1.5
and 3
3
8
6. Hence x = 6 feet. This gives an area of 324 ft2.
1.5
9
2. The side of the square is √18 inches. Hence the area is 18π square inches.
3. Let x be
and y be
. Then x = 9y and 180
9 180
. Solving gives
18°
4. Translating 4 units right and then 4 units down gives 9 units. The diagonal translation sweeps an area of
6 additional units. The “diagonal” sweep is a trapezoid made from an isosceles right triangle with legs
length 4 units minus an isosceles right triangle with legs length 2 units.
136
5.
6. The sum of the interior angles of a regular octagon is 8*135° = 1080°. The sum of the exterior angles of
a regular octagon is 360°. Hence the difference is 720°.
7. √55.86
3.9
56 m
8. Let 2x be the side of the triangle. Then
9. Let x be the corresponding side. Then
√
√
and
√
. Hence
√
√
√
0.5
solving gives x = 11.2 inches.
10. Let E be the point on side AC such that line DE is parallel to side AB. It is clear that triangle ABC is
similar to triangle EDC. Also triangle ADE is an isosceles triangle because angle BAD is congruent to
angle ADE by alternate interior angles of parallel lines, but angle BAD is congruent to angle DAE by
the definition of angle bisector. Using the two pieces of information it can be shown that AE = DE = 2
cm. Furthermore using similarity again it can be shown that AB = 6 cm.
11. 64
18
183 square inches.
12. There are seven cubes of volume one cubic foot – the original cube plus the cubes on each of the six
faces. There are three cylinders of height one foot and radius one foot – a fourth of a cylinder on each
of the twelve edges. Finally there is one sphere of radius one foot – and eighth of a sphere at each of the
eight vertices. Adding these all up gives approximately 21 cubic feet.
12.5. Solving for x gives 0.5 feet.
13. Let x be the depth of the water. Then 10
14. The largest possible area is a rectangle of whose sides are 4 segments by 5 segments or 10 inches by
12.5 inches. Thus the largest area is 125 square inches.
15. The number of tiles in one diagonal must be 19. Hence the total number of tiles is 192 = 361.
16.
2.
by the Pythagorean Theorem so
8
2011 State Math Contest
Wake Technical Community College
17. 3
2
2
3
Geometry Test
3
3
180. Hence y = 36°. If 3
2 , then x = 24°. If 3
3 , then x = 12°. Hence
60° or
54° or
3
3 , then x = 18°. If
48°.
18. Sides PA and QB are both perpendicular to side AB. Let T be a point on side PA such that segment QT
is perpendicular to segment PA. PTQ is a right triangle with sides 6 inches, 12inches, and 6√3 inches.
Hence the quadrilateral APQB has an area of 36√3 square inches.
19. Let s be the side of cube A. Let r be the side of cube B. Then 6
this gives
.7155 or 72%.
360°
20.
1.25 6
or
√0.8. Hence
90.467° or 90° and 28 minutes.
21.
22. Let x be the length of side RT, 2x be the length of side RS, and y be the side of the square REDF.
Triangle SED is similar to triangle SRT, hence
. Solving gives x = 1.5y. Also
triangle SED is similar to triangle DFT, hence
2.
.
23. Point B is the point 4,1 and point C is the point 1, 4 . The area of the triangle is 15 square units.
The easiest way to see this is to draw the rectangle through 1, 4 ; 1,4 ; 4,4 ; and 4, 4 . This
rectangle has an area of 40 square units, then subtract off the area of the three extra triangles.
24. Let
and
0.5 and
. Then
1.25 . Thus
1.5 . Hence
0.75
.
25. Assume all three letters chosen are different then there are 210 different strings. If the two E’s are
chosen then there are six ways to choose the additional letter and 3 ways to arrange the 3 letters. Hence
an additional 18 strings. This gives 228 strings.
66. Let x be the length of the radius, then by Heron’s Theorem the area of the triangle is 42 16 12 14
or 336 square centimeters. The triangle can be split into two triangles of height x and bases of 30 and
26. Hence, 15
13
336 so x = 12 cm.
67. Let x be the length and y be the width of the prism. Then 8
8
2
4
or
. Using the
table feature of the calculator the solution is 8 inches by 8 inches.
68. The radius of the circle of intersection of the plane and the sphere can be found using the Pythagorean
Theorem to be √12 feet, hence the area is 37.7 square feet to the nearest tenth of a square foot.
69. The length of CD is feet and the length of DB is feet. Hence the side of the triangle ABC is
√
√
√
feet. That makes the height of the triangle 6 feet.
70. The length of CK is √9
2.25
√11.25 feet. Hence the length of BK is √11.25
9
9 feet or 4.5 feet.