Question

1. this diagram shows the design for a quilt. the quilt is made of 4 squares that have been divided along their diagonals. the length of segment bd is 4. a. find the length of segment ae. b. find the area of square abcd.

Explanation:

Step1: Relate AE to BD

Since the diagonals of a square bisect each other and are equal, $AE = \frac{1}{2}BD$.
Expression: $AE = \frac{1}{2} \times 4$

Step2: Calculate AE length

Compute the value of $\frac{1}{2} \times 4$.
Expression: $AE = 2$

Step3: Find side length of square

Let side length of square $ABCD$ be $s$. For a square, diagonal $BD = s\sqrt{2}$, so $s = \frac{BD}{\sqrt{2}}$.
Expression: $s = \frac{4}{\sqrt{2}} = 2\sqrt{2}$

Step4: Calculate area of square

Area of square is $s^2$. Substitute $s = 2\sqrt{2}$.
Expression: $\text{Area} = (2\sqrt{2})^2 = 8$

Answer:

a. $2$
b. $8$