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Math 11
A monster of a unit 2 review
Short Answer
1. Determine the measure of ∠ABF.
2. Determine the values of a, b, and c.
3. Determine the values of a, b, and c.
4. Determine the measure of ∠PQR.
5. Given QP || MR, determine the measure of ∠QMO.
6. Determine the value of x.
7. Each interior angle of a regular convex polygon measures 156°.
How many sides does the polygon have?
8. Gareth is measuring the exterior angles of a convex hexagon.
So far, he has measured 60°, 60°, 60°, 30°, and 30°.
What is the measure of the last exterior angle?
Show your calculation.
9. Sketch a triangle that corresponds to the equation.
Then, determine the third angle measure and the third side length.
10. Determine the length of d to the nearest tenth of a centimetre.
11. Determine the measure of θ to the nearest degree.
12. Determine the length of w to the nearest tenth of a centimetre.
13. Determine the measure of θ to the nearest degree.
14. A kayak leaves a dock on Lake Athabasca, and heads due north for 2.8 km. At the same time, a second kayak
travels in a direction N70°E from the dock for 3.0 km.
How you can determine the distance between the kayaks?
15. A radar operator on a ship discovers a large sunken vessel lying parallel to the ocean surface, 180 m directly
below the ship. The length of the vessel is a clue to which wreck has been found. The radar operator measures
the angles of depression to the front and back of the sunken vessel to be 52° and 67°. How long, to the nearest
tenth of a metre, is the sunken vessel?
16. Determine the unknown angle measure to the nearest degree.
17. In ∆ABC, ∠A = 26°, a = 8.5 cm, and b = 5.0 cm. Determine the number of triangles (zero, one, or two) that
are possible for these measurements. Draw the triangle(s) to support your answer.
18. In ∆ABC, ∠A = 45°, a = 6.0 cm, and b = 7.5 cm. Determine the number of triangles (zero, one, or two) that
are possible for these measurements. Draw the triangle(s) to support your answer.
19. In ∆UVW, ∠V = 73° and VW = 18.6 cm. Calculate the height of the triangle from base VU to the nearest tenth
of a centimetre.
20. Determine the indicated angle measure to the nearest degree.
Problem
21. Do you need to know QP || MR to determine the measure of ∠QMO? Explain.
22. Prove: FG || HI
23. A floor tiler designs custom floors using tiles in the shape of regular polygons. The tiler uses three different
tile shapes to cover a floor, all with the same side length. At each corner, there is one square and one hexagon.
What is the third tile shape? Draw part of the tiling.
24. A radio tower is supported by two wires on opposite sides. On the ground,
the ends of the wire are 280 m apart. One wire makes a 60° angle with the ground.
The other makes a 66° angle with the ground.
Draw a diagram of the situation. Then, determine the length of each wire to the nearest metre. Show your
work.
25. Stella decided to ski to a friend’s cabin. She skied 8.0 km in the direction N40°E. She rested, then skied
S30°E and arrived at the cabin. The cabin is 9.5 km from her home, as the crow flies. Determine, to the
nearest tenth of a kilometre, the distance she travelled on the second leg of her trip. Show your work.
26. The pendulum of a grandfather clock is 85.0 cm long. When the pendulum
swings from one side to the other side, it travels a horizontal distance of 10.5 cm.
Determine the angle through which the pendulum swings. Round your answer to the nearest tenth of a degree.
27. While golfing, Beth hits a tee shot from point T toward a hole at H. However, the ball veers 34° and lands at B.
The scorecard says that H is 250 m from T. Beth walks 120 m to her ball. Sketch a diagram of this situation.
How far, to the nearest metre, is her ball from the hole? Show your work.
28. The posts of a hockey goal are 2.0 m apart. A player is standing at a point 4.5 m from one post and 6.0 m from
the other post. Within what angle must the player shoot the puck to score a goal? Express your answer to the
nearest degree. Show your work.
29. A building is observed from two points, P and Q, that are 94.0 m apart. The angle of elevation is 42° at P and
33° at Q. Sketch the situation. Determine the height of the building to the nearest tenth of a metre.
U2 review
Answer Section
SHORT ANSWER
1. ANS:
∠ABF = 66°
2. ANS:
∠a = 18°, ∠b = 54°, ∠c = 27°
3. ANS:
∠a = 15°, ∠b = 30°, ∠c = 10°
4. ANS:
∠PQR = 122°
5. ANS:
∠QMO = 23°
6. ANS:
x = 48°
7. ANS:
15
8. ANS:
360° – 60° – 60° – 60° – 30° – 30° = 120°
9. ANS:
70°, 18.8
10. ANS:
d = 6.2 cm
11. ANS:
θ = 57°
12. ANS:
w = 27.3 cm
13. ANS:
θ = 57°
14. ANS:
Since the measures of two sides and a contained angle are given, I would use the cosine law.
15. ANS:
217.0 m
16. ANS:
42°
17. ANS:
one triangle:
18. ANS:
two triangles:
19. ANS:
17.8 cm
20. ANS:
49° or 131°
PROBLEM
21. ANS:
No. Use alternate interior angles and complementary angles to determine ∠MNO. Use the sum of angles in a
triangle to determine ∠NMO. Then use the sum of angles on a straight line to solve for ∠QMO.
22. ANS:
∠FHG + ∠GHI + ∠IHJ
94° + ∠GHI + 73°
∠GHI
∠GHI
∠GHI
Therefore,
FG
= 180°
= 180°
= 180° – 94° – 73°
= 13°
= ∠FGH
|| HI
Sum of angles in triangle is 180°
Substitute known values.
Determine ∠GHI.
equal alternate interior angles
23. ANS:
The measure of an interior angle of a square is 90°.
The measure of an interior angle of a regular hexagon is 120°.
This leaves a gap of 360° – 90° – 120° = 150°.
Determine the number of sides, n, of a regular polygon with 150°-angles:
The measure of the interior angles in a regular 12-sided polygon (dodecagon) is 150°.
The tiling is made with a square, a regular hexagon, and a regular dodecagon:
24. ANS:
Let the x and y be the length of the wires.
The third angle is 180° – 66° – 60° = 54°.
Use the sine law to determine the length of each wire:
The wires are 316 m and 300 m long.
25. ANS:
Because the lines are parallel, the angle beside the 30° angle is also 40°.
The entire angle is 70°.
x + 70° + z = 180°
x + 70° + 52.309...° = 180°
x = 57.690...°
Stella travelled 8.5 km.
26. ANS:
a2
10.52
110.25
–14 339.75
= b2 + c2 – 2bc cos A
= 85.02 + 85.02 – 2(85.0)(85.0) cos A
= 7225.00 + 7225.00 – 14 450.00 cos A
= –14 450.00 cos A
= cos A
∠A = cos–1(0.9923...)
∠A = 7.082...°
The pendulum swings through an angle of 7.1°.
27. ANS:
By the cosine law,
t2 = h2 + b2 – 2hb cos T
t2 = 1202 + 2502 – 2(120)(250) cos 34°
t2 = 27 157.745...
t = 164.796...
Beth's ball is 165 m from the hole.
28. ANS:
Draw a diagram of the situation.
By the cosine law,
a2
2.02
2
2
2.0 – 4.5 – 6.02
–52.25
= b2 + c2 – 2bc cos C
= 4.52 + 6.02 – 2(4.5)(6.0) cos C
= – 2(4.5)(6.0) cos C
= –54 cos C
= cos C
cos–1
= ∠C
14.6264…° = ∠C
The player must shoot within a 15° angle.
29. ANS:
The measures of ∠PRQ, PQ, and ∠PQR are known. Use the sine law to determine PR.
PR is 327.3 m.
The measures of ∠RSP, PR, and ∠SPR are known. Use the sine law to determine RS, or h, the height of the
building.
The height of the building is 219.0 m.