Question

if the ratio of the shortest and longest sides of a right angled triangle be 3:5 and its perimeter is 36cm then find the area of the triangle.

Answer 1

Given:-

• Ratio of the shortest and longest side of the right-angled triangle = 3:5
• Perimeter = 36 cm

To Find:-

• Area of the traingle

Note:-

• Refer to the attachment for the diagram.

Solution:-

Let the ratio common be x

Hence the ratio becomes:- 3x and 5 x

Let us assume the shorter side to be the base i.e., AB

Hence, AB = 3x

And as we know the largest side of a right-angled triangle is always hypotenuse.

Hence, BC = 5x

Now,

By using the Pythagoras theorem we can find the third side of the traingle.

Hence,

By using Pythagoras Theorem,

(BC)² = (AB)² + (AC)²

=> (AC)² = (BC)² - (AB)²

=> $\sf{AC = \sqrt{(5x)^2 - (3x)^2}}$

=> $\sf{AC = \sqrt{25x^2 - 9x^2}}$

=> $\sf{AC = \sqrt{16x^2}}$

=> $\sf{AC = 4x}$

Now,

Perimeter = Sum of all the three sides

Hence,

36 = 3x + 4x + 5x

=> 36 = 12x

=> x = $\sf{\dfrac{36}{12}}$

=> x = 3 cm

Putting the value of x in the sides of the triangle,

AB = 3x = 3 × 3 = 9 cm

BC = 5x = 5 × 3 = 15 cm

AC = 4x = 4 × 3 = 12 cm

Now,

We know,

Area of a right-angled triangle = $\sf{\dfrac{1}{2}\times base\times height}$

Hence,

$\sf{Area = \dfrac{1}{2}\times 9\times 4}$

= $\sf{Area = 18\:cm^2}$

Hence the area of the triangle is 18 cm².

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