Question

a garden is designed in the shape of a rhombus formed from 4 identical 30° - 60° - 90° triangles. the shorter distance across the middle of the garden measures 30 feet. what is the distance around the perimeter of the garden? 60 ft 60√3 ft 120 ft 120√3 ft

Explanation:

Step1: Recall the property of a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, if the shorter leg (opposite the \(30^{\circ}\) angle) is \(x\), the hypotenuse is \(2x\). Here, the shorter leg of each right - angled triangle (half of the shorter diagonal of the rhombus) is \(15\) ft.

Step2: Find the side length of the rhombus

Let the side length of the rhombus be \(s\). Using the \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle ratio, if the shorter leg \(a = 15\) ft (in the right - angled triangle formed by the diagonals of the rhombus), then the hypotenuse (side of the rhombus) \(s=2\times15 = 30\) ft.

Step3: Calculate the perimeter of the rhombus

The perimeter \(P\) of a rhombus is given by \(P = 4s\). Since \(s = 30\) ft, then \(P=4\times30=120\) ft.

Answer:

\(120\) ft (the third option)