Geometry Individual 2017 Mu Alpha Theta

Complete each of the following problems and place your answers on the answer document provided.

1. If the clock below reads 3:08, what is the measure of the acute angle formed between the two hands?

2. In the figure below, ∠BAC measures 30°, ∠ABC measures 110°, and points B, C, and D are collinear.

What is the measure of ∠ACD?

3. In the isosceles right triangle below, AB = 10 feet. What is the length, in feet, of AC? Leave in square

root form.

4. In the circle shown below, M is the center and lies on RU and ST. The measure of ∠RMS is 50º and the

radius is 10 units. What is the measure of the actual arc length of RT? Round to 2 decimals.

5. What is the maximum number of distinct diagonals that can be drawn in a regular hexagon?

Geometry Individual 2017 Mu Alpha Theta

Complete each of the following problems and place your answers on the answer document provided.

6. In the right triangle below, the measure of ∠C is 90º, AB = 5 units, and CB = 2 units. What is tan B ?

7. In ΔABD, C is on BD, the length of AD is 6 inches, and sin d = .8. How many inches long is CD?

8. What would be the number of meters a runner must run in a 3-lap race where the course has a circular

path of radius R meters?

9. If DE = 2X + 4, EF = 6X – 10, and EF = 2DE. Find the measure of DF.

•--------------•-----------------------------•

D E F

10. In the 8-sided figure below, adjacent sides meet at right angles and the lengths given are in meters.

What is the perimeter of the figure in meters?

11. The parallel sides of the isosceles trapezoid shown below are 10 feet and 16 feet, respectively. What is

the distance, in feet, between these 2 sides?

Geometry Individual 2017 Mu Alpha Theta

Complete each of the following problems and place your answers on the answer document provided.

12. The youth center has installed a swimming pool on level ground. The pool is a right circular cylinder

with a diameter of 24 feet and a height of 6 feet. A diagram of the pool with its entry ladder is shown

below. To the nearest cubic foot, what is the volume of water that will be in the pool when it is filled

with water to a depth of 5 feet?

13. A plastic cover is made for the same pool mentioned above in problem 11. The cover will rest on the

top of the pool and will include a wedge-shaped flap that forms a 45º angle at the center of the cover, as

shown below. A zipper will go along 1 side of the wedge-shaped flap and around the arc. What is the

length, in feet, of the zipper?

14. In the figure below, line l is parallel to line m. Transversals t and u intersect at point A on line l and

intersects line m at points C and B. Point X is on line m, the measure of ∠ACX is 130°, and the measure of ∠BAC is 80°. How many angles formed by rays l, m, t, and u have a measure of 50°?

15. In the figure below, 2 nonadjacent sides of a regular pentagon are extended until they meet at point X.

measure of ∠X? What is the

Geometry Individual 2017 Mu Alpha Theta

Complete each of the following problems and place your answers on the answer document provided.

16. In the figure below, where ΔABC ∼ΔKLM, lengths given are in centimeters. What is the perimeter of

ΔABC ?

17. In the figure below, the area of the larger square is 50 square centimeters and the area of the smaller

square is 18 square centimeters. What is the measure of x in centimeters?

18. In the circle shown below, chords TR and QS intersect at P, which is the center of the circle, and the

measure of ∠PST is 30°. What is the degree measure of minor arc RS?

19. The radius on the base of the right circular cone is 5 in, and the height of the cone is 7 in. If the cone’s base is sitting flat on the ground, how much material would be needed to cover the entire top of the cone?

Geometry Individual 2017 Mu Alpha Theta

Complete each of the following problems and place your answers on the answer document provided.

20. Calculate the scale factor of the dilation around the center C. The preimage is on the interior of the

image.

21. A rectangular wall of height H feet and width L feet is to be painted. If the wall has 4 rectangular

windows, each of height X feet and width W feet, what is the number of square feet to be painted?

22. In the figure below, AD is a line segment whose length is 35 units. The lengths of AC and BD are 14

and 27 units respectively. What is the length of BC in units?

•-----------•-----------•----------------------•

A B C D

23. What is the complement of the supplement of 155° ?

24. If each exterior angle of a regular n-gon has a measure of 12°, what is n? (number of sides)

25. The vertices of a polygon are given as W(-2,2), X(0,2), Y(4,-2), and Z(-6.-2). What is the area of this

polygon?

Geometry Individual 2017 Mu Alpha Theta Solutions

1. The angle between any 2 minute marks is 6°. There are 7 minutes between the two hands therefore 7 x 6° = 42°

2. If angle A and B are 30° and 110° respectively then there sum gives you the remote exterior angle ACD which is

140°.

3. 45-45-90 triangle hypotenuse is time the leg therefor 10 or 14.14 ft.

4. If arc length is found by the fractional part of the circle times the circumference then we use the formula

2πr(

) which is 2(π)(10)(

) which results in 22.69 or 7.22π.

5. 9 distinct diagonals

6. Find AC first using Pythagorean Theorem = AC = therefore tan B = op/adj =

or 2.29

7. Sin d = .8 which is 8/10. Using the number on the triangle for sin d = x / 6. Therefore 8/10 = x/6 which x = 4.8. To

find CD use Pythagorean Theorem = CD = 3.6 in.

8. Circumference is 2πR, so if the runner runs 3 laps then 3(2πR) = 6πR or 18.85R

9. EF = 2DE therefore 6x-10 = 2(2x+4) which makes x = 22 and DF = 66.

10. If all the concave parts are pushed out, then a perfect square of 20 x 20 is formed. Therefore pereimeter is 4(20)

= 80.

11. The distance between the two parallel sides would be the height of the trapezoid. Using Pythagorean Theorem

you can solve for the height to get 4. Therefore the distance between the two parallel sides is 4 ft.

12. Volume of Cylinder = π h = π ( 5) = 2261.95 or 2262

13. The arc length can be found by multiplying the fractional part of the circle of the arc times the circumference.

(2πr) =

(2)( π)(12)=9.42 This is then added to the radius of 12 which results in 12 + 9.42 = 21.42

feet or 3π+12 ft

14. Because of linear pair, 180° – 130° = 50° for angle ACB. Then by using vertical angles and alternate interior with

the parallel lines you can fill in the 50° in the following angles:

15. Interior Angle Sum of Pentagon is 540°. 540° / 5 = 108° for each angle. (b/c regular pentagon) Linear pair makes

the 2 congruent exterior angles form the extended sides 72°. Therefore with triangle sum 180° – 144° = 36°

16. Using similar triangle ratios we can determine that AB = 5 and BC = 6. Therefore the 3+5+6 = 14.

17. Larger square area is 50 therefore X(X)=50 making the larger side . The smaller square area is 18 therefore

X(X) = 18 making the smaller side . Subtract the smaller side from the larder side resulting in - =

or 2.83 cm

18. The 30° angle is an inscribed angle that makes its arc 60°. The central angle is the same as the arc making it 60°.

Vertical angles then repeats the process on the opposite side making RS = 60°

19. Cone Surface Area = area of base + πrl ( radius and lateral or slant height). Since the base is flat on the ground

and not covered, we only need the πrl part. π (5)(8.6) = 135.13 or 5π (the 8.6 comes from using

Pythagorean Theorem with the right triangle inside the cone.

20. The distance from the center of the image to any vertex on the image is 3x the distance of the center of the

preimage to any corresponding vertex on the preimage.

21. LH – 4XW (area of wall minus the area of the 4 windows)

22. AD – AC = CD = 35 – 14 = 21 AD – BD = AB = 35 – 27 = 8 AD – (CD + AB) = BC = 35-(21+8) = 6

23. 180 – 155 = 25 90 – 25 = 65

24. Exterior angle sum of any regular polygon is 360°. Therefore 360° / 12° = 30 sides or 30-gon.

25. Plotting the points results in the following trapezoid with bases of 2 and 10 and a height of 4.

The area of a trapezoid is

=

= 24