Question

The perimeter of a rhombus is 52 cm, if its one diagonal is 24 cm then the length of its other diagonal is

• A. 5 cm
• B. 7 cm
• C. 9 cm
• D. 10 cm

Answer 1

The Correct Option is D

To solve the problem, we need to find the length of the other diagonal of a rhombus given that its perimeter is \(52 \, \text{cm}\) and one of its diagonals is \(24 \, \text{cm}\).

Step 1: Understand the properties of a rhombus.
A rhombus has all sides of equal length. The diagonals of a rhombus bisect each other at right angles (90°). This means that each diagonal divides the rhombus into four right-angled triangles.

Step 2: Calculate the side length of the rhombus.
The perimeter of the rhombus is given as \(52 \, \text{cm}\). Since a rhombus has four equal sides, the length of each side (\(s\)) is:

\[ s = \frac{\text{Perimeter}}{4} = \frac{52}{4} = 13 \, \text{cm} \]

Step 3: Use the properties of the diagonals.
Let the diagonals of the rhombus be \(d_1\) and \(d_2\), where \(d_1 = 24 \, \text{cm}\) and \(d_2\) is the unknown diagonal. The diagonals bisect each other at right angles, so each half of \(d_1\) is \(12 \, \text{cm}\) and each half of \(d_2\) is \(\frac{d_2}{2}\).

In one of the right-angled triangles formed by the diagonals, the hypotenuse is the side of the rhombus (\(s = 13 \, \text{cm}\)), and the legs are half the lengths of the diagonals (\(12 \, \text{cm}\) and \(\frac{d_2}{2}\)). By the Pythagorean theorem:

\[ s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]

Substitute the known values (\(s = 13\) and \(\frac{d_1}{2} = 12\)):

\[ 13^2 = 12^2 + \left(\frac{d_2}{2}\right)^2 \]

Simplify:

\[ 169 = 144 + \left(\frac{d_2}{2}\right)^2 \]

Solve for \(\left(\frac{d_2}{2}\right)^2\):

\[ \left(\frac{d_2}{2}\right)^2 = 169 - 144 = 25 \]

Take the square root of both sides:

\[ \frac{d_2}{2} = \sqrt{25} = 5 \]

Multiply both sides by 2 to find \(d_2\):

\[ d_2 = 2 \times 5 = 10 \, \text{cm} \]

Final Answer:
The length of the other diagonal is \(10 \, \text{cm}\).

\[ {4} \]