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 properties of a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\). If the shorter side (opposite the \(30^{\circ}\) angle) is \(x\), the hypotenuse is \(2x\).

Step2: Find the length of the side of the rhombus

The shorter diagonal of the rhombus is \(30\) feet. When the rhombus is divided into four \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles, the shorter side of each triangle (opposite the \(30^{\circ}\) angle) is \(15\) feet.
For a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, if the side opposite \(30^{\circ}\) is \(15\) feet, then the hypotenuse (which is the side of the rhombus) \(s = 2\times15=30\) feet.

Step3: Calculate the perimeter of the rhombus

The perimeter \(P\) of a rhombus is \(4s\) (since all sides of a rhombus are equal). Substituting \(s = 30\) feet, we get \(P=4\times30 = 120\) feet.

Answer:

\(120\) ft