Geometry 8-6 Complete Lesson: Law of Cosines - Matthew Richardson | Laabri

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.

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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.

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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.

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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)

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Problem 1 Got It?

A

B

C

D

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Problem 2 Got It?

A

B

C

D

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Problem 3 Got It?

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B

C

D

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A

B

C

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A

B

C

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B

C

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A

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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?

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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.

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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

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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

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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

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

45°
100°
90°
61°
110°
179°
2°

Acute
Obtuse
Right

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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

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Reflection: Math Success

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