Question

1. in rectangle abcd, diagonals ac and bd intersect at point e. if ae = 36 and bd = 2x + 30, find x

Explanation:

Step1: Recall rectangle diagonal property

In a rectangle, diagonals are equal and bisect each other. So, \( AC = BD \) and \( AE=\frac{1}{2}AC \), \( BE = \frac{1}{2}BD \), which implies \( AC = 2AE \).
Given \( AE = 36 \), then \( AC=2\times36 = 72 \).

Step2: Set up equation for BD

Since \( AC = BD \) (diagonals of rectangle are equal), and \( BD = 2x + 30 \), \( AC = 72 \), we have the equation:
\( 2x + 30=72 \)

Step3: Solve for x

Subtract 30 from both sides: \( 2x=72 - 30=42 \)
Divide both sides by 2: \( x=\frac{42}{2}=21 \)

Answer:

\( x = 21 \)