Question

applying the 30°-60°-90° and 45°-45°-90° relationships
isaiah sketches a model of a skateboard ramp. the model has two surfaces on which to skate, represented by sides ab and ad in the diagram.
the steepest side of the model, ab, measures 4 inches. what is the length of the other skating surface, ad?
options: (2sqrt{2}) in., (2sqrt{3}) in., (4sqrt{2}) in., (4sqrt{3}) in.
diagram: triangle with a at top, b and d at base, c is foot of perpendicular from a to bd. angle at b is 45°, angle at a between ab and ac is 45°, angle at a between ac and ad is 60°, angle at d is 30°, right angle at c.

Explanation:

Step1: Find length of AC (shared leg)

In isosceles right triangle ABC ($45^\circ$-$45^\circ$-$90^\circ$), the leg is $\frac{\text{hypotenuse}}{\sqrt{2}}$.
$AC = \frac{AB}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}$ inches

Step2: Find length of AD (hypotenuse)

In $30^\circ$-$60^\circ$-$90^\circ$ triangle ACD, the hypotenuse is $2 \times$ shorter leg (AC is the shorter leg opposite $30^\circ$).
$AD = 2 \times AC = 2 \times 2\sqrt{2} = 4\sqrt{2}$ inches

Answer:

$4\sqrt{2}$ in.