Question

A right angled triangle having perimeter 120 cm has its two perpendicular sides in the ratio
5:12. Find the lengths of its sides.​

Answer 1

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Answer 2

The length of the sides of a right angled triangle are :

• The first side = 20 cm.
• The second side = 48 cm.
• The third side = 52 cm.

Given :

• The perimeter of a right angled triangle = 120 cm.
• The ratio of the two perpendicular sides = 5 : 12.

To Find :

• The length of the sides of a right angled triangle.

Solution :

Let,

The first of the triangle (base) be 5x.

The second side of the triangle (perpendicular) be 12x.

First, we need to find the third side (hypotenuse).

By Pythagoras theorem,

$\bullet \rm \: \: {a}^{2} = {b}^{2} + {c}^{2}$

Where,

• a = hypotenuse.
• b = base = 5x.
• c = perpendicular = 12x.

$: \implies \rm {a}^{2} = {(5x)}^{2} + {(12x)}^{2} \\ \\ : \implies \rm {a}^{2} = 25 {x}^{2} + 144 {x}^{2} \\ \\ : \implies \rm {a}^{2} = 169 {x}^{2} \\ \\ : \implies \rm a = \sqrt{169 {x}^{2} } \\ \\ : \implies \rm a = 13x \\ \\ \: \: \therefore \: \: \sf hypotenuse = 13x$

Now, we have to find the length of the sides of a right angled triangle.

Given,

• Perimeter of the triangle = 120 cm

We know that,

• Perimeter of the triangle = a + b + c

That means,

$: \implies \rm a + b + c = 120 \: cm$

We have,

• a = 13x.
• b = 5x.
• c = 12x.

$: \implies \rm 13x + 5x + 12x = 120 \: cm \\ \\ : \implies \rm 13x + 17x = 120 \: cm \\ \\ : \implies \rm 30x = 120 \: cm \\ \\ : \implies \rm x = \dfrac{120}{30} \: cm \\ \\ : \implies \rm x = \dfrac{12}{3} \: cm \\ \\ : \implies \rm x = 4 \: cm \\ \\ \: \: \therefore \: \: \rm x = 4 \: cm$

So the length of the sides of a right angled triangle are :

★ The first side = 5x = 5 × 4 cm = 20 cm.

★ The second side = 12x = 12 × 4 cm = 48 cm.

★ The third side = 13x = 13 × 4 cm = 52 cm.

Hence,

The length of the sides of a right angled triangle are 20 cm, 48 cm and 52 cm.