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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