G.SRT.D.10 WORKSHEET – geometrycommoncore 2

Unit 7.5 Lesson 3 HW

1. Which of the following three pieces of information work with the Law of Sines?

/ a) Yes or No / b) Yes or No / c) Yes or No
mÐA = 56°
b = 17 cm
a = 24 cm / mÐB = 107°
b = 15 cm
a = 9 cm / mÐA = 47°
mÐB = 98°
a = 24 cm
d) Yes or No / e) Yes or No / f) Yes or No
mÐA = 49°
mÐC = 33°
b = 9 cm / a = 10 cm
b = 15 cm
c = 9 cm / mÐC = 41°
b = 14 cm
a = 24 cm
g) Yes or No / h) Yes or No / i) Yes or No
mÐA = 39°
a = 8 cm
b = 9 cm / mÐA = 20°
mÐB = 65°
mÐC = 95° / mÐC = 41°
mÐB = 98°
a = 15 cm

2. Using the data in question #2, state the order that the information comes in and then state whether it is a guaranteed congruence relationship or not.

a) AS1S2 S2 > S1 / b) ______/ c) ______/ d) ______/ e) ______
Congruence?
Yes / Congruence?
Yes or Not / Congruence?
Yes or Not / Congruence?
Yes or Not / Congruence?
Yes or Not
f) ______/ g) ______/ h) ______/ i) ______
Congruence?
Yes or Not / Congruence?
Yes or Not / Congruence?
Yes or Not / Congruence?
Yes or Not

3. Explain why the Law of Sines doesn’t work for DABC if you are given, mÐA, b and c.

4. Jonathan says that you can’t use the Law of Sines in DABC if you are given mÐA, mÐB and c because there is no ‘pairing’ of an angle with its opposite side. Brittney disagrees with Jonathan. Brittney is correct; you can use the Law of Sines with this situation. Explain why it is possible.

5. AS1S2 while not always producing a congruence relationship does work in the Law of Sines. Why does AS1S2 always guarantee a ‘pairing’ of an angle and its opposite side? Draw a diagram to help explain your answer.

6. One of the cases of AS1S2 can produce two different triangles. Explain when and why it happens.

7. Solve for all the sides and angles of DABC using the Law of Sines.

a) / b)
/ mÐA = 38°
mÐB = 96°
mÐC = ______
a = 9 cm
b = ______
c = ______
/ / mÐA = ______
mÐB = 82°
mÐC = 59°
a = ______
b = ______
c = 23 cm
c) / d)
/ mÐA = 40°
mÐB = ______
mÐC = 70°
a = ______
b = ______
c = 12 cm
/ / mÐA = 27°
mÐB = ______
mÐC = 121°
a = ______
b = 34 cm
c = ______

8. Draw a diagram, and then solve for all the sides and angles of DABC using the Law of Sines,
given that mÐA = 11°, mÐB = 85° and a = 45 m.

mÐA = 11° a = 45 m
mÐB = 85° b = ______m
mÐC = ______c = ______m

9. Solve the following problems.

a) The tallest tree in the forest is 150 feet from the water tower. Unfortunately in a lightening storm the tallest tree was struck and caught on fire. The ranger needs to go from his tower to the water tower to get the water to put out the fire. How far is it from the Ranger’s Tower to the Water Tower? (nearest foot)
b) A surveyor is able to gather some information about the terrain around a river. How far is it across the river from A to B? (to the nearest metre)
c) A surveyor in a boat on the river at point A was able to gather some information about the terrain around a river. How far is it across the river (from B to C to the nearest foot), if the angle from the surveyor to the two points on the shore (ÐBAC) was 13° and the angle from the shore at point B to the boat and the other point on the shore (ÐABC) is 97° and the estimated distance from A to B is 75 ft?

10. Explain the four cases of AS1S2 by drawing a diagram and explaining what takes place.

CASE #1 - AS1S2 where S2 > S1 / CASE #2 - AS1S2 where S2 < S1 (No intersection)
Diagram / Diagram
CASE #3 - AS1S2 where S2 < S1 (One intersection) / CASE #4 - AS1S2 where S2 < S1 (Two intersections)
Diagram / Diagram

11. Identify what you can expect from the given data. If you are guaranteed 1 solution select that, otherwise choose 0/1/2 solutions.

a) mÐA = 53°, a = 15 cm, b = 10 cm 1 solution or 0/1/2 solutions

b) mÐA = 53°, mÐB = 111°, b = 10 cm 1 solution or 0/1/2 solutions

c) mÐB = 25°, a = 15 cm, mÐC = 76° 1 solution or 0/1/2 solutions

d) mÐC = 43°, c = 12 cm, a = 15 cm 1 solution or 0/1/2 solutions

e) mÐB = 65°, a = 11 cm, b = 12 cm 1 solution or 0/1/2 solutions

f) mÐC = 53°, mÐA = 77°, b = 23 cm 1 solution or 0/1/2 solutions

g) a = 30 cm, c = 34 cm, mÐC = 23° 1 solution or 0/1/2 solutions

h) mÐA = 61°, a = 9 cm, b = 10 cm 1 solution or 0/1/2 solutions

12. Determine if the given three items could be used in the Law of Sines.

a) mÐA = 32°, a = 13 cm, b = 10 cm Sine Law or Not Sine Law

b) a = 13 cm, b = 11 cm, c = 12 cm Sine Law or Not Sine Law

c) mÐA = 55°, a = 13 cm, mÐB = 39° Sine Law or Not Sine Law

d) mÐA = 12°, b = 45 cm, mÐC = 47° Sine Law or Not Sine Law

e) mÐC = 68°, a = 13 cm, b = 10 cm Sine Law or Not Sine Law

13. To understand the different case of AS1S2 we will do one of each type of problem. This will help you understand the different cases.

a) mÐA = 73°, a = 13 cm, b = 10 cm
Draw a Diagram / What kind of solution do you expect from this data?
Solve the triangle for all angles and sides
mÐA = 73°
mÐB = ______
mÐC = ______
a = 13 cm
b = 10 cm
c = ______cm
b) mÐA = 45°, a = 10 cm, c = cm
Draw a Diagram / What kind of solution do you expect from this data?
Solve the triangle for all angles and sides
mÐA = 45°
mÐB = ______
mÐC = ______
a = 10 cm
b = ______cm
c = cm
c) mÐC = 55°, a = 23 cm, c = 15 cm
Draw a Diagram / What kind of solution do you expect from this data?
Solve the triangle for all angles and sides
mÐA = ______
mÐB = ______
mÐC = 55°
a = 23 cm
b = ______cm
c = 15 cm
d) mÐA = 56°, a = 18 cm, c = 20 cm
Draw a Diagram / What kind of solution do you expect from this data?
Solve the triangle for all angles and sides
mÐA = 56°
mÐB = ______
mÐC = ______
a = 18 cm
b = ______cm
c = 20 cm
mÐA = 56°
mÐB = ______
mÐC = ______
a = 18 cm
b = ______cm
c = 20 cm
14. Solve the triangle for all angles and sides. / mÐA = 60°, b = 16 cm, a = cm
Draw a Diagram / mÐA = 60°
mÐB = ______
mÐC = ______
a = cm
b = 16 cm
c = ______cm
(If needed)
mÐA = 60°
mÐB = ______
mÐC = ______
a = cm
b = 16 cm
c = ______cm
15. Solve the triangle for all angles and sides. / mÐA = 28°, a = 18 cm, c = 25 cm
Draw a Diagram / mÐA = 28°
mÐB = ______
mÐC = ______
a = 18 cm
b = ______cm
c = 25 cm
(If needed)
mÐA = 28°
mÐB = ______
mÐC = ______
a = 18 cm
b = ______cm
c = 25 cm
16. Solve the triangle for all angles and sides. / mÐC = 70°, a = 18 cm, c = 15 cm
Draw a Diagram / mÐA = ______
mÐB = ______
mÐC = 70°
a = 18 cm
b = ______cm
c = 15 cm
(If needed)
mÐA = ______
mÐB = ______
mÐC = 70°
a = 18 cm
b = ______cm
c = 15 cm
17. Solve the triangle for all angles and sides. / mÐA = 41°, a = 9 cm, c = 6 cm
Draw a Diagram / mÐA = 41°
mÐB = ______
mÐC = ______
a = 9 cm
b = ______cm
c = 6 cm
(If needed)
mÐA = 41°
mÐB = ______
mÐC = ______
a = 9 cm
b = ______cm
c = 6 cm
18. Solve the triangle for all angles and sides. / mÐA = 25°, a = 10 cm, c = 20 cm
Draw a Diagram / mÐA = 25°
mÐB = ______
mÐC = ______
a = 10 cm
b = ______cm
c = 20 cm
(If needed)
mÐA = 25°
mÐB = ______
mÐC = ______
a = 10 cm
b = ______cm
c = 20 cm

19. Why doesn’t SSS and SAS information work with the Law of Sines?

20. Jennifer states that if the cosine ratio of an angle in a triangle is negative then it is an obtuse angle. She is correct. Why would the cosine of an obtuse angle be negative? /
21. Using the Law of Cosines, Jeremy solves for side b. The teacher warns Jeremy to solve for the smallest angle next.
a) Which angle is the smallest? ______
b) How can you determine which angle is smaller of the two?
c) Why would the teacher warn Jeremy to do this? /
22. Jazmine is about to solve this triangle by using the Law of Cosines. The teacher warns Jazmine to solve for the largest angle first.
a) Which angle is the largest? ______
b) How can you determine which angle is the largest?
c) Why would the teacher warn Jazmine to do this? /

23. It keeps coming up that sin Ɵ = sin (180 – Ɵ), explain why that is true.

24. Why did we need the Law of Cosines? Why wasn’t the Law of Sines good enough?

25. Write out the three versions of the Law of Cosines for the given triangle.

26. Solve the triangle by determining all of its sides and angles.

mÐA = 28°
mÐB = ______
mÐC = ______
a = ______cm
b = 18 cm
c = 11 cm

27. Solve the triangle by determining all of its sides and angles.

mÐA = ______
mÐB = ______
mÐC = ______
a = 17 cm
b = 19 cm
c = 15 cm

28. Given DABC, where mÐB = 34°, a = 8 cm and c = 12 cm. Solve the triangle.

mÐA = ______
mÐB = 34°
mÐC = ______
a = 8 cm
b = ______cm
c = 12 cm

29. Solve the following problems.

a) A forest ranger spots a tree that is on fire. He is able to determine that he is 210 ft from the fire and is 288 ft from the water tower. If the angle between the tower and the fire is 57°, how far is the water tower from the fire? (nearest ft)
b) A surveyor is able to gather some information about the terrain around a river. He plans to build a bridge to the island from reference point A and from reference point B. What is the angle formed at the island between those two reference points (ÐBCA)? (nearest degree)
c) A rock climber at Point A stops to analyze the terrain around the river. From his high vantage point he is able to determine that he is 125 ft from west shore (point B) and 200 ft from the east shore (point C) and the angle between those two locations is 25°. What is the width of the river? (nearest foot)