Q: Given the triangle in the second image, which formula would you use to solve for $$a$$?

$$a^2$$ = $$800^2$$ + $$700^2 - 2$$ \times 800 \times 700 \cos 75^\circ$$

Q: Which of the following is Heron's formula for the area of a triangle with sides $$a$$, $$b$$, and $$c$$?

$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$ where $$s = \frac{a+b+c}{2}$$

Question 1

Given the triangle in the first image, what is the value of angle $$C in $$degrees?

Accepted Answer

$$C = 84^\circ$$

Question 2

Using the triangle in the first image, what is the value of $$a ($$the distance from B to C) to the nearest tenth of a mile?

Accepted Answer

$$a \approx 10.7$$ miles

Question 3

Which law is most appropriate to solve for side $$a in $$the first triangle?

Accepted Answer

Law of Sines

Question 4

Given the triangle in the second image, which formula would you use to solve for $$a$$?

Accepted Answer

$$a^2$$ = $$800^2$$ + $$700^2 - 2$$ 	imes 800 	imes 700 \cos 75^\circ$$

Question 5

Calculate the length of $$a in $$the second triangle to the nearest mile.

Accepted Answer

$$a \approx 917$$ miles

Question 6

A triangle has sides $$a = 6 m$$, $$b = 16 m$$, and $$c = 18 m. $$What is the area of the triangle to the nearest square meter?

Accepted Answer

$$47 m$^2$$

Question 7

Which of the following is Heron's formula for the area of a triangle with sides $$a$$, $$b$$, and $$c$$?

Accepted Answer

$$	ext{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$ where $$s = \frac{a+b+c}{2}$$

Question 8

A surveyor measures two sides of a triangular plot as $$20 ft $$and $$40 ft$$, with an included angle of $$135^\circ. $$What is the area of the plot to the nearest square foot?

Accepted Answer

$$297 ft$^2$$

Question 9

Given triangle $$ABC$$ with $$A = 35^\circ$$, $$a = 12$$, $$b = 16$$, how many possible triangles can be formed?

Accepted Answer

Two triangles

Question 10

If $$a = 10$$, $$b = 20$$, and $$A = 50^\circ$$, what can you conclude about the triangle?

Accepted Answer

No triangle exists

Question 11

Which scenario best describes when to use the Law of Cosines instead of the Law of Sines?

Accepted Answer

When you know all three sides (SSS) or two sides and the included angle (SAS)

Question 12

A triangle has sides $$a = 8$$, $$b = 10$$, $$c = 5. $$What is the measure of angle $$B to $$the nearest degree?

Accepted Answer

$$98^\circ$$

Question 13

Given triangle $$ABC$$ with $$A = 120^\circ$$, $$b = 7$$, $$c = 8$$, what is the length of side $$a$$?

Accepted Answer

$$a = 13$$

Question 14

A triangle has sides $$a = 11$$, $$b = 9$$, $$c = 7. $$What is the area to the nearest square yard?

Accepted Answer

$$31 yd$^2$$

Question 15

Two fire-lookout stations are 10 miles apart, with station B directly east of station A. The bearing of the fire from station A is $$N25^\circ E$$ and from station B is $$N56^\circ W. $$How far is the fire from station B to the nearest tenth of a mile?

Accepted Answer

$$9.2$$ miles

Chapter 7: Additional Topics in Trigonometry – Law of Sines and Law of Cosines

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