Question

The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is
144 cm. Find the area of the triangle.
​

Answer 1

Step-by-step explanation:

$\frak Given = \begin{cases} &\sf{The\ sides\ of\ the\ triangle\ are\ in\ the\ ratio\ 3\ :\ 4\ :\ 5.} \\ &\sf{Perimeter\ of\ the\ triangle\ is\ 144cm.} \end{cases}$

To find:- We have to find the area of the triangle ?

☯️ Let the sides of triangle are 3x, 4x, and 5x. And the perimeter of the triangle is 144 cm.

So here:-

$\sf : \implies {3x\ +\ 4x\ +\ 5x\ =\ 144} \\ \\ \sf : \implies {12x\ =\ 144} \\ \\ \sf : \implies {x\ =\ \cancel \dfrac{144}{12}} \\ \\ \sf : \implies {\purple{\underline{\boxed{\bf x\ =\ 12.}}}}\bigstar$

$\frak{\underline{\underline{\dag So\ the\ sides\ of\ the\ triangle:-}}}$

$\sf : \implies {3x\ =\ 3\ ×\ 12\ =\ 36cm.}$

$\sf : \implies {4x\ =\ 4\ ×\ 12\ =\ 48cm.}$

$\sf : \implies {5x\ =\ 5\ ×\ 12\ =\ 60cm.}$

__________________

$\frak{\underline{\underline{\dag As\ we\ know\ that:-}}}$

$\sf : \implies {\pink{\underline{Area_{(triangle)}\ =\ \sqrt{s(s-a)(s-b)(s-c)}}}}$

● Here s = 72, a = 36, b = 48, and c = 60.

___________________

$\frak{\underline{\underline{\dag By\ substituting\ the\ values,\ we\ get:-}}}$

$\sf : \implies {\sqrt{72(72-36)(72-48)(72-60)}} \\ \\ \sf : \implies {\sqrt{72(36)(24)(12)}} \\ \\ \sf : \implies {\sqrt{746496}} \\ \\ \sf : \implies {\purple{\underline{\boxed{\bf Area\ =\ 864cm².}}}}\bigstar$

Hence:-

$\sf \therefore {\underline{The\ required\ area\ of\ the\ triangle\ is\ 864cm².}}$

Answer 2

Given :

• Sides of the triangle = 3 : 4 : 5
• Perimeter = 144 cm

To find :

• The area of the triangle

Solution :

★ Let:-

• The sides of the triangle = 3x, 4x, 5x

•°• 3x + 4x + 5x = 144

→ 12x = 144

→ x = 144/12

→ x = 12

★ Hence:-

• The sides of the triangle = 36, 48, 60

Now we will use Heron's Fomula to find out the area of the triangle. First we will find the semicircle. Then we will find the area of the triangle using the formula.

• Semicircle = 1/2(36 + 48 + 60)

→ 1/2(144)

→ 72

• Area = √{s(s - 36)(s - 48)(s - 60)}

→ √{72(72 - 36)(72 - 48)(72 - 60)}

→ √(72 × 36 × 24 × 12)

→ √746496

→ 864

•°• the area of the triangle = 864 m²

Answer :

• The area of the triangle is 864 m²