Question

1. if de = 4x + 1, eb = 12x - 31, and cd = 28, find ad.

Explanation:

Step1: Set segments equal (rectangle diagonals bisect)

$4x + 1 = 12x - 31$

Step2: Solve for x

$31 + 1 = 12x - 4x$
$32 = 8x$
$x = \frac{32}{8} = 4$

Step3: Calculate diagonal DB length

First find $DE = 4(4) + 1 = 17$, so $DB = 2 \times DE = 2 \times 17 = 34$

Step4: Use Pythagorean theorem for AD

In rectangle, $AD^2 + CD^2 = DB^2$, so $AD = \sqrt{DB^2 - CD^2}$
$AD = \sqrt{34^2 - 28^2} = \sqrt{(34-28)(34+28)} = \sqrt{6 \times 62} = \sqrt{372} = 2\sqrt{93} \approx 19.29$

Answer:

$2\sqrt{93}$ (or approximately 19.29)