Question 1

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

Accepted Answer

Heron's formula for the area of a triangle is given by $\sqrt{s(s-a)(s-b)(s-c)}$, where $s$ is the semi-perimeter of the triangle, calculated as $s = \frac{a+b+c}{2}$.

Question 2

What is the semi-perimeter ($s$) of a triangle with sides $10	ext{ cm}$, $24	ext{ cm}$, and $26	ext{ cm}$?

Accepted Answer

The semi-perimeter $s$ is calculated as $\frac{a+b+c}{2} = \frac{10+24+26}{2} = \frac{60}{2} = 30	ext{ cm}$.

Question 3

Find the area of an equilateral triangle with side $4	ext{ cm}$.

Accepted Answer

For an equilateral triangle with side $a = 4$, the area is $\frac{\sqrt{3}}{4}a^2 = \frac{\sqrt{3}}{4}(4)^2 = 4\sqrt{3}	ext{ cm}^2$.

Question 4

The sides of a triangle are in the ratio $12:17:25$ and its perimeter is $540	ext{ cm}$. What is its longest side?

Accepted Answer

Let the sides be $12x$, $17x$, and $25x$. Perimeter = $12x + 17x + 25x = 54x = 540$, so $x = 10$. The longest side is $25x = 25 	imes 10 = 250	ext{ cm}$.

Question 5

What is the area of a triangle with sides $3	ext{ cm}$, $4	ext{ cm}$, and $5	ext{ cm}$?

Accepted Answer

The sides form a right-angled triangle since $3^2 + 4^2 = 5^2$. Area = $\frac{1}{2} 	imes 	ext{base} 	imes 	ext{height} = \frac{1}{2} 	imes 3 	imes 4 = 6	ext{ cm}^2$.

Heron's Formula