Question

If the length of the diagonal and the area of a rectangle are 25 m and 168 m\(^2\), respectively, then the longer side of the rectangle is:

• A. 21 m
• B. 22 m
• C. 24 m
• D. 20 m

Hint:

When dealing with rectangles and their properties, using Pythagoras’ theorem for the diagonal and the area equation can help you solve for the unknown sides.

Answer 1

The Correct Option is C

Let the length of the rectangle be \( l \) and the breadth be \( b \). We are given the following: - The length of the diagonal is 25 m. By Pythagoras’ theorem, we know: \[ l^2 + b^2 = 25^2 = 625 \] - The area of the rectangle is 168 m\(^2\), so: \[ l \times b = 168 \] Now, we have two equations: 1. \( l^2 + b^2 = 625 \) 2. \( l \times b = 168 \) To solve for \( l \) and \( b \), we use substitution or the quadratic method. Let’s substitute \( l = \frac{168}{b} \) into the first equation: \[ \left( \frac{168}{b} \right)^2 + b^2 = 625 \] Simplifying: \[ \frac{28224}{b^2} + b^2 = 625 \] Multiply through by \( b^2 \): \[ 28224 + b^4 = 625b^2 \] Let \( x = b^2 \), then: \[ 28224 + x^2 = 625x \] Rearranging: \[ x^2 - 625x + 28224 = 0 \] Now solving this quadratic equation for \( x \) gives \( x = 24 \). Therefore, \( b = 24 \). So, the longer side of the rectangle is \( \boxed{24 \, \text{m}} \).