Question

Find the area of the triangle with sides 21 cm, 16 cm and 13 cm. Also, find the perimeter of an equilateral triangle equal in area to this triangle

Answer 1

$\bf{Answer}$

Step-by-step explanation:

sum of sides = 21cm+16cm+13cm = 50cm = s

s/2 = 25

By Heron's formula:

$\begin{gathered}\sqrt{(s)(s-a)(s-b)(s-c)}\\= \sqrt{(25)(25-21)(25-16)(25-13)}\\= \sqrt{(25)(4)(9)(12)}\\\\= 5*2*3 *2\sqrt{3}\\= 60\sqrt{3} cm^2\\\end{gathered}$

Area of equilateral triangle: = $\frac{\sqrt{3}}{4}a^2$

$\begin{gathered}\frac{\sqrt{3}}{4}a^2 = 60\sqrt{3}\\a^2 = 240\\a = 4\sqrt{15}\\\end{gathered}$

Perimeter:

$\begin{gathered}=3a\\= 3 * 4\sqrt{15}\\=12 \sqrt{15}cm\end{gathered}$

Answer 2

Here, as per the provided question we are asked to find the perimeter of an equilateral triangle which is equal in area to this triangle –

GIVEN :

• The area of the triangle with sides 21cm, 16cm and 13cm are given respectively.

TO FIND :

• The perimeter of an equilateral triangle is equal in area to this triangle = ?

STEP-BY-STEP EXPLANATION :

Now, we will find the perimeter of an equilateral triangle here –

Formula used :

→ Area of the triangle (heron's formula)

→ Side = a + b + c/2

[ substituting the values as per formula ] :

→ Side = 21 + 16 + 13 / 2

→ side = 50/2 [ When reduced] = 25

→ area = √s (s-a) × (s-b) × (s-c)

→ area = √25 (25-21) × (25-16) × (25-13)

→ area = √25 × 4 × 9 × 12

→ area = √5 × 5 × 2 × 2 × 3 × 3 × 2 × 3 × 2

→ area = 60√3

Now, we got here area = 60√3 .

Therefore, perimeter of an equilateral triangle = ?

→ perimeter = √3/4 (a²)

→ 60√3 = √3/4 (a²)

→ 60 = (a²)/4

→ a² = 4 × 60

→ a = √240

→ a = 4√15

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

→ area = 3 × 4√15

→ area = 12√15

Hence, perimeter of an equilateral triangle = 12√15 .