Question

The lengths of the sides of the right triangle are in the ratio 3:4:5. What is the area of the triangle if the perimeter is 24 cm?

Answer 1

Answer:

480

Step-by-step explanation:

the ratio is 3:4:5

so, we can write 3x , 4x , 5x

perimeter of triangle=sum of all sides

24 = 3x+4x+5x

24 = 12x

24/12=X

2=X

all sides measure

3x=3×2=6

4x=4×2=8

5x=5×2=10

area = sss

= 6×8×10

= 480cm

Answer 2

• Area of the triangle = 24 cm²

Given :–

• The ratio of the lengths of the sides of the right triangle = 3 : 4 : 5.
• Perimeter of the triangle = 24 cm.

To Find :–

• The area of the triangle.

How to solve ?

First we need to find all the three sides of the triangle which is given in the form of ratio of 3 : 4 : 5 by using the given value of the perimeter of the triangle. After finding all the three sides of the triangle we apply the heron's formula for getting the value of area of the triangle.

Heron's formula = $\sf \sqrt{s(s-a)(s-b)(s-c)}$ where s is semi-perimeter.

For find the value of s (semi perimeter).

We use,

$\sf s = \dfrac{a + b + c}{2}$ and a, b and c is the three sides of the triangle.

Solution :–

Let,

The first side of the triangle be 3x.

The second side of the triangle be 4x.

The third side of the triangle be 5x.

First, we need to find the sides of the triangle.

Given that,

Perimeter of the triangle = 24 cm.

$\mapsto 3x + 4x + 5x = 24 \: cm$

$\mapsto 7x + 5x = 24 \: cm$

$\mapsto 12x = 24 \: cm$

$\mapsto x = \cancel\dfrac{24}{12} \: cm$

$\mapsto x = 2 \: cm$

So, the sides are :

First side = 3x = 3 × 2 = 6 cm.

Second side = 4x = 4 × 2 = 8 cm.

Third side = 5x = 5 × 2 = 10 cm.

Now, we have to find the area of the triangle.

By heron's formula,

$\mapsto \sqrt{s(s - a)(s - b)(s - c)}$

Where,

$\mapsto s = \dfrac{a + b + c}{2}$

We have,

• a = 6 cm.
• b = 8 cm.
• c = 10 cm.

$\mapsto s = \dfrac{6 + 8 + 10}{2} \: cm$

$\mapsto s = \dfrac{14 + 10}{2} \: cm$

$\mapsto s = \cancel\dfrac{24}{2} \: cm$

$\mapsto s = 12 \: cm$

Now we have,

• s = 12 cm.
• a = 6 cm.
• b = 8 cm.
• c = 10 cm.

Now, substitute all the values in the heron's formula.

$\mapsto \sqrt{12(12 - 6)(12 - 8)(12 - 10) \: {cm}^{4} }$

$\mapsto \sqrt{12(6)(4)(2)} \: {cm}^{2}$

$\mapsto \sqrt{(2 \times 2 \times 3)(2 \times 3)(2 \times 2)(2)} \: {cm}^{2}$

$\mapsto \sqrt{2 \times 2 \times 3 \times 2 \times 3 \times 2 \times 2 \times 2} \: {cm}^{2}$

$\mapsto \sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3} \: {cm}^{2}$

$\mapsto \sqrt{ {(2)}^{2} \times {(2)}^{2} \times {(2)}^{2} \times {(3)}^{2} } \: {cm}^{2}$

$\mapsto 2 \times 2 \times 2 \times 3 \: {cm}^{2}$

$\mapsto 4 \times 6 \: {cm}^{2}$

$\mapsto 24 \: {cm}^{2}$

Hence,

The area of the triangle is 24 cm².