Q: Which formula is used to find the area of a triangle given two sides and the included angle?

$$\text{Area} = \frac{1}{2}ab\sin C$$

Q: Which formula is used to find the area of a triangle when all three sides are known?

$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$

Question 1

Which equation represents the Law of Sines for a triangle with angles $$A$$, $$B$$, $$C$$ and sides $$a$$, $$b$$, $$c$$ opposite those angles?

Accepted Answer

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Question 2

A triangle has sides $$a = 10$$, $$b = 12$$, and $$c = 14. $$What is the area of the triangle using Heron's Formula?

Accepted Answer

$$58.8$$

Question 3

Given triangle $$ABC$$ with $$b = 12$$, $$c = 8$$, and angle $$C = 25^	ext{o}$$, which law should you use to find side $$a$$?

Accepted Answer

Law of Cosines

Question 4

In the ambiguous case (SSA), which scenario can result in two possible triangles?

Accepted Answer

When the given angle is acute and the side opposite is shorter than the other given side but longer than the altitude.

Question 5

Which formula is used to find the area of a triangle given two sides and the included angle?

Accepted Answer

$$	ext{Area} = \frac{1}{2}ab\sin C$$

Question 6

A road slopes $$15^	ext{o}$$ above the horizontal, and a vertical telephone pole stands beside the road. The angle of elevation of the Sun is $$65^	ext{o}$$, and the pole casts a $$15-$$foot shadow downhill along the road. What is the height of the pole?

Accepted Answer

$$27.2$$ feet

Question 7

Which equation represents the Law of Cosines for side $$a$$?

Accepted Answer

$$a^2$$ = $$b^2$$ + $$c^2 - 2bc$$\cos A$$

Question 8

If you know two angles and one side of a triangle (AAS or ASA), which law is most appropriate to solve for the unknown sides?

Accepted Answer

Law of Sines

Question 9

Given triangle $$ABC$$ with $$a = 7$$, $$b = 9$$, and $$C = 40^	ext{o}$$, what is the first step to determine if the ambiguous case applies?

Accepted Answer

Calculate the height from $$C to$$ $$AB$$ and compare it to $$a$$

Question 10

Which of the following is NOT a valid use of the Law of Sines?

Accepted Answer

Solving a triangle given all three sides (SSS)

Question 11

A triangle has sides $$a = 8$$, $$b = 15$$, and $$c = 17. $$What is the semiperimeter $$s$$?

Accepted Answer

$$s = 20$$

Question 12

Which scenario best illustrates the ambiguous case (SSA) in real life?

Accepted Answer

A surveyor measures two sides of a plot and the angle opposite one side, but finds two possible locations for the third corner.

Question 13

If a triangle has sides $$a = 5$$, $$b = 7$$, and $$c = 10$$, what is the area using Heron's Formula?

Accepted Answer

$$16.25$$

Question 14

Which of the following is a correct application of the Law of Cosines to find angle $$A$$?

Accepted Answer

$$\cos A = \frac{b^2$ + c^2$ - a^2$}{2bc}$$

Question 15

A triangle has sides $$a = 6$$, $$b = 8$$, and angle $$C = 30^	ext{o}. $$What is the area of the triangle?

Accepted Answer

$$24$$

Question 16

Which of the following is true about the sum of the angles in any triangle?

Accepted Answer

$$180^	ext{o}$$

Question 17

If you are given $$a = 10$$, $$b = 12$$, and $$C = 45^	ext{o}$$, which law would you use to find angle $$A$$?

Accepted Answer

Law of Sines

Question 18

Which formula is used to find the area of a triangle when all three sides are known?

Accepted Answer

$$	ext{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$

The Laws of Sines and Cosines: Solving Triangles and Applications

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