Three sides of a right-angled triangle are in the ratio of 7 : 24 : 25. If the perimeter of the triangle is 112 cm, then length of the hypotenuse is equal to :

Question

Accepted Answer

To determine the length of the hypotenuse of the right-angled triangle, we start by understanding the given information:

The sides of the triangle are in the ratio 7:24:25. This ratio indicates that the triangle is a right-angled triangle, where 25 is the hypotenuse because it is the largest number in the ratio, and in a right-angled triangle, the hypotenuse is always the longest side.

Given that the perimeter of the triangle is 112 cm, we can set up an equation to find the actual lengths of the sides. Let the common multiple of the sides be $ x $. Therefore, the sides of the triangle can be expressed as:
- First side = $ 7x $
- Second side = $ 24x $
- Hypotenuse = $ 25x $

The perimeter of the triangle is the sum of its sides:
$$ 7x + 24x + 25x = 112 $$

Simplifying the equation:
$$ 56x = 112 $$

Solving for $ x $:
$$ x = \frac{112}{56} = 2 $$

Now, we can find the actual lengths of the sides:
- First side = $ 7 \times 2 = 14 $ cm
- Second side = $ 24 \times 2 = 48 $ cm
- Hypotenuse = $ 25 \times 2 = 50 $ cm

Thus, the length of the hypotenuse is 50 cm, which corresponds to Option 3.

**Explanation of why other options are incorrect:**

1. **Option 1 (56 cm):** This option is incorrect because, based on the ratio and the perimeter, the calculated hypotenuse is 50 cm, not 56 cm.

2. **Option 2 (48 cm):** This option is incorrect because 48 cm corresponds to one of the other sides of the triangle, not the hypotenuse.

3. **Option 4 (14 cm):** This option is incorrect because 14 cm is the length of the smallest side of the triangle, not the hypotenuse.

Therefore, the correct answer is Option 3, where the length of the hypotenuse is 50 cm.