Answer :
Answer:
Step-by-step explanation:
Let's denote the lengths of the two perpendicular sides of the right-angled triangle as \(5x\) and \(12x\), respectively. According to the Pythagorean theorem, the hypotenuse \(c\) of the triangle can be expressed as:
\[
c = \sqrt{(5x)^2 + (12x)^2} = \sqrt{25x^2 + 144x^2} = \sqrt{169x^2} = 13x
\]
We are given that the perimeter of the triangle is 120 cm. The perimeter \(P\) of the triangle is the sum of its three sides:
\[
P = 5x + 12x + 13x = 30x
\]
Setting the perimeter equal to 120 cm, we get:
\[
30x = 120
\]
Solving for \(x\):
\[
x = \frac{120}{30} = 4
\]
Now, we can find the lengths of the sides by substituting \(x = 4\) back into the expressions for the sides:
\[
\text{First perpendicular side} = 5x = 5 \times 4 = 20 \text{ cm}
\]
\[
\text{Second perpendicular side} = 12x = 12 \times 4 = 48 \text{ cm}
\]
\[
\text{Hypotenuse} = 13x = 13 \times 4 = 52 \text{ cm}
\]
Thus, the lengths of the sides of the triangle are 20 cm, 48 cm, and 52 cm.
To verify, we check the perimeter and the Pythagorean theorem:
\[
\text{Perimeter check:} \quad 20 + 48 + 52 = 120 \text{ cm}
\]
\[
\text{Pythagorean theorem check:} \quad 20^2 + 48^2 = 400 + 2304 = 2704 \quad \text{and} \quad 52^2 = 2704
\]
Both checks confirm that the calculations are correct. Therefore, the lengths of the sides of the triangle are indeed 20 cm, 48 cm, and 52 cm.