Question

the longest side of the triangle when the ratio of 3:4:5. if it's perimeter is 36, what is is area
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Answer 1

$Answer \:$

Formula :

Perimeter of the triangle = a + b + c

Solution :

Let the sides of the triangle are 3x,4x and 5x.

A/C: 3x+4x+5x=36

➠ 12x = 36

➠ x = 36/12

➠ x = 3

Thus, 3x = 9cm, 4x = 12cm and 5x = 15cm

Area :

1/2 × base × height

$➠ \frac{1}{2} \times 9 \times 12 \\ ➠1 \times 6 \times 9 \\ ➠54 {cm}^{2}$

Therefore, The area of the triangle is 54cm².

Answer 2

Given:-

• The longest side of the triangle in the ratio of 3:4:5.
• Perimeter of the triangle is 36 cm.

⠀

To find:-

• Area of the triangle.

⠀

Solution:-

Let,

• the ratio be x.

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {AB + BC + CA = 36}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {3x + 4x + 5x = 36}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {12x = 36}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {x = \dfrac{36}{12}}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {x = 3}$

⠀

• Putting the value of x.

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {AB = 3x = 9\: cm}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {BC = 4x = 12\: cm}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {AC = 5x = 15\: cm}$

⠀

${\dag}\:{\underline{\boxed{\sf{\purple{Using\: Pythagoras\: theorem}}}}}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {AC^2 = AB^2 + BC^2}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {(9)^2 + (12)^2}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {81 + 144}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {AC^2 = 225}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {AC = 15\: cm}$

⠀

• ∠B is right angle in ΔABC.

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {\triangle ABC = \dfrac{1}{2} AB \times BC}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {\dfrac{1}{2} 9 \times 12}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {\underline{\boxed{\pink{54\: cm^2}}}}$

⠀

Hence,

• the area of the triangle is 54 cm².