Question

prom the prom committee is decorating the venue for prom and wants to hang lights above the diagonals of the rectangular room. if $dh$=44.5 feet, $ef$=39 feet, and $m\angle ghf$=128°, find $dg$.
$dg=$ ______ feet

Explanation:

Step1: Recall rectangle diagonal properties

In a rectangle, diagonals are equal and bisect each other. So $DF = EG$, and $DH = HF = GH = HE$.

Step2: Calculate length of diagonal $EG$

Since $DH = 44.5$ feet, $EG = 2 \times DH = 2 \times 44.5 = 89$ feet.

Step3: Use Pythagorean theorem for $\triangle DEG$

$DG$ is a side of the rectangle, $EF = DG = 39$ feet is not needed here; instead, for right triangle $\triangle DEG$, $EG^2 = DG^2 + DE^2$, but we know $EF = DE = 39$ feet (opposite sides of rectangle are equal). Rearrange to solve for $DG$:
$$DG = \sqrt{EG^2 - DE^2}$$
Substitute $EG = 89$, $DE = 39$:
$$DG = \sqrt{89^2 - 39^2}$$

Step4: Compute the squares

$89^2 = 7921$, $39^2 = 1521$
$$DG = \sqrt{7921 - 1521} = \sqrt{6400}$$

Step5: Simplify the square root

$\sqrt{6400} = 80$

Answer:

80 feet