a garden is designed in the shape of a rhombus formed from 4 identical 30 - 60 - 90 triangles. the shorter…

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

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

Answer

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 triangle, the side lengths are in the ratio $1:\sqrt{3}:2$. The shorter leg (opposite the 30 - degree angle) is given as half of the shorter diagonal of the rhombus. Here, the shorter diagonal is 30 feet, so the shorter leg of each 30 - 60 - 90 triangle is 15 feet.

Step2: Find the side length of the rhombus

The side length of the rhombus is equal to the hypotenuse of the 30 - 60 - 90 triangle. Since the shorter leg ($a = 15$ feet) and the hypotenuse $c$ of a 30 - 60 - 90 triangle satisfy $c = 2a$, the side length of the rhombus $s=30$ feet.

Step3: Calculate the perimeter of the rhombus

The perimeter $P$ of a rhombus is given by $P = 4s$. Substituting $s = 30$ feet, we get $P=4\times30=120$ feet.

Answer:

120 ft