Geometry 8-6 Law of Cosines

by Matthew Richardson

| 20 QuestionsNote from the author:

A complete formative lesson with embedded slideshow, mini lecture screencasts, checks for understanding, practice items, mixed review, and reflection. I create these assignments to supplement each lesson of Pearson's Common Core Edition Algebra 1, Algebra 2, and Geometry courses. See also mathquest.net and twitter.com/mathquestEDU.

Sovle It! In the diagram below, △ABC is an acute triangle. Use what you know about right triangle trigonometry to write an expression for the area of the shaded region that uses a, b, and c.

1

10 pts

Solve It! Step 1: Consider the smaller right triangle. Write an equation relating its unknown leg length (also the width of the orange rectangle) with b and C.

2

10 pts

Solve It! Step 2: Solve the equation from the previous item for x.

x = b cos(C)

x = C cos(b)

x = b sin(C)

x = tan(bC)

We can see that the area of the shaded region = ax, but the instructions are to write an expression for the area of the shaded region that uses a, b, and c.

3

10 pts

Solve It! Step 3: Use the Substitution Property of Equality and the Commutative Property of Multiplication to find the equation that satisfies the Solve It!.

Area of shaded rectangle = aC cos(b)

Area of shaded rectangle = ab cos(C)

Area of shaded rectangle = aC sin(b)

Area of shaded rectangle = ab sin(C)

4

4

10 pts

Problem 1 Got It?

A

B

C

D

5

5

10 pts

Problem 2 Got It?

A

B

C

D

6

6

10 pts

Problem 3 Got It?

A

B

C

D

7

7

10 pts

A

B

C

D

8

8

10 pts

A

B

C

D

9

9

10 pts

A

B

C

D

10

10

10 pts

A

B

C

D

11

11

10 pts

Error Analysis: In △ABC, AC = 15 ft, BC = 12 ft, and m∠C = 32. A student solved for c for a = 12 ft, b = 15 ft, and m∠C = 32. What was the error?

13

10 pts

Review Lesson 8-4: The first leg of a bike race is 6 km due east. For the second leg of the race, the riders turn northwest and ride 9 km. The final leg of the race runs at a 40° angle back to the second leg and brings the racers back to the starting point. To the nearest tenth, at what angle measure does the third leg of the race meet the first leg? Enter only a number.

14

5 pts

Review Lesson 6-7: Is the quadrilateral with vertices A(-1, -5), B(6, -5), C(9, 3), and D(2, 3) a parallelogram?

Yes

No

16

10 pts

Review Lesson 4-1: In the diagram above, △ABC ≅ △EFG. Use this information to match congruent parts from the triangles.

- segment AB
- segment FG
- ∠E
- ∠C
- segment EG

- segment BC
- ∠G
- segment EF
- segment AC
- ∠A

17

10 pts

Vocabulary Review: Match the parts from the triangle above in the left column with the descriptions in the right column. Not all parts of the triangle will be used.

- a
- b
- c
- A
- B
- C

- Sides adjacent to angle A
- Side opposite angle B
- Sides adjacent to angle C

18

10 pts

Vocabulary Review: Classify each angle as acute, obtuse, or right.

- 45°
- 100°
- 90°
- 61°
- 110°
- 179°
- 2°

- Acute
- Obtuse
- Right

19

10 pts

Use Your Vocabulary: Identify each ratio as representing sine, cosine, tangent, or none of these.

- hypotenuse/adjacent
- adjacent/hypotenuse
- opposite/hypotenuse
- adjacent/opposite
- opposite/adjacent
- hypotenuse/opposite

- Sine
- Cosine
- Tangent
- None of these

20

10 pts

Reflection: Math Success

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