Q: Which formula correctly gives the area of a triangle when two sides $$a$$ and $$b$$ and the included angle $$C$$ are known?

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

Q: A triangle has sides $$a = 7$$ units, $$b = 8$$ units, and included angle $$C = 60^\\circ. $$What is its area?

$$\dfrac{1}{2} \times 7 \times 8 \times \sin 60^\\circ = 24.25$$ $$units^2$

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 $$s = \dfrac{1}{2}(a+b+c)$$

Q: Using Heron's formula, what is the area of a triangle with sides $$a = 5$$ units, $$b = 6$$ units, and $$c = 7$$ units?

$$\sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} \approx 14.7$$ $$units^2$

Q: Which formula would you use to find the area of a triangle if you know two sides and the angle between them?

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

Q: A triangle has sides $$a = 8$$ units, $$b = 15$$ units, and included angle $$C = 90^\\circ. $$What is its area?

$$\dfrac{1}{2} \times 8 \times 15 \times \sin 90^\\circ = 60$$ $$units^2$

Q: A triangle has angles $$A$$, $$B$$, and $$C$$, and sides $$a$$, $$b$$, and $$c$$ opposite those angles. Which formula gives the area in terms of $$a$$ and $$B$$?

$$\text{Area} = \dfrac{1}{2} a c \sin B$$

Q: A triangle has sides $$a = 9$$ units, $$b = 12$$ units, and included angle $$C = 45^\\circ. $$What is the area?

$$\dfrac{1}{2} \times 9 \times 12 \times \sin 45^\\circ = 38.18$$ $$units^2$

Question 1

Given a triangle with base $$b = 10$$ units and height $$h = 6$$ units, what is the area of the triangle?

Accepted Answer

$$30$$ $$units^2$

Question 2

Which formula correctly gives the area of a triangle when two sides $$a$$ and $$b$$ and the included angle $$C$$ are known?

Accepted Answer

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

Question 3

A triangle has sides $$a = 7$$ units, $$b = 8$$ units, and included angle $$C = 60^\circ. $$What is its area?

Accepted Answer

$$\dfrac{1}{2} 	imes 7 	imes 8 	imes \sin 60^\circ = 24.25$$ $$units^2$

Question 4

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 $$s = \dfrac{1}{2}(a+b+c)$$

Question 5

A triangle has sides $$a = 5$$ units, $$b = 6$$ units, and $$c = 7$$ units. What is the value of the semi-perimeter $$s$$?

Accepted Answer

$$s = 9$$

Question 6

Using Heron's formula, what is the area of a triangle with sides $$a = 5$$ units, $$b = 6$$ units, and $$c = 7$$ units?

Accepted Answer

$$\sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 	imes 4 	imes 3 	imes 2} = \sqrt{216} \approx 14.7$$ $$units^2$

Question 7

If a triangle has sides $$a = 10$$ units, $$b = 12$$ units, and $$c = 14$$ units, which of the following is the correct expression for its area using Heron's formula?

Accepted Answer

$$\sqrt{18(18-10)(18-12)(18-14)}$$

Question 8

A surveyor needs to find the area of a triangular plot with sides $$a = 50 m$$, $$b = 60 m$$, and $$c = 70 m. $$What is the first step in applying Heron's formula?

Accepted Answer

Calculate the semi-perimeter $$s = \dfrac{1}{2}(a+b+c)$$

Question 9

Which formula would you use to find the area of a triangle if you know two sides and the angle between them?

Accepted Answer

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

Question 10

A triangle has sides $$a = 8$$ units, $$b = 15$$ units, and included angle $$C = 90^\circ. $$What is its area?

Accepted Answer

$$\dfrac{1}{2} 	imes 8 	imes 15 	imes \sin 90^\circ = 60$$ $$units^2$

Question 11

Which of the following is NOT a valid formula for the area of a triangle?

Accepted Answer

$$	ext{Area} = ab + bc + ca$$

Question 12

A triangle has angles $$A$$, $$B$$, and $$C$$, and sides $$a$$, $$b$$, and $$c$$ opposite those angles. Which formula gives the area in terms of $$a$$ and $$B$$?

Accepted Answer

$$	ext{Area} = \dfrac{1}{2} a c \sin B$$

Question 13

A triangle has sides $$a = 9$$ units, $$b = 12$$ units, and included angle $$C = 45^\circ. $$What is the area?

Accepted Answer

$$\dfrac{1}{2} 	imes 9 	imes 12 	imes \sin 45^\circ = 38.18$$ $$units^2$

Question 14

Which of the following is the correct formula for the area of triangle $$ABC in $$terms of $$a$$, $$b$$, and $$C$$?

Accepted Answer

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

Question 15

A triangle has sides $$a = 13$$ units, $$b = 14$$ units, and $$c = 15$$ units. What is the area using Heron's formula?

Accepted Answer

$$\sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 	imes 8 	imes 7 	imes 6} = \sqrt{7056} \approx 84$$ $$units^2$

Question 16

A civil engineer is designing a triangular park with sides $$a = 30 m$$, $$b = 40 m$$, and $$c = 50 m. $$What is the area of the park?

Accepted Answer

$$\sqrt{60(60-30)(60-40)(60-50)} = \sqrt{60 	imes 30 	imes 20 	imes 10} = \sqrt{360000} = 600$$ $$m^2$

Question 17

Which formula would you use to find the area of a triangle if you know all three sides?

Accepted Answer

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

Question 18

A triangle has sides $$a = 11$$ units, $$b = 13$$ units, and $$c = 15$$ units. What is the semi-perimeter $$s$$?

Accepted Answer

$$s = 19$$

Area of Triangles: Trigonometric Methods and Applications

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