A garden is designed in the shape of a rhombus formed from 4 identical [tex]$30^{\circ}-60^{\circ}-90^{\circ}$[/tex] triangles. The shorter distance across the middle of the garden measures 30 feet.
What is the distance around the perimeter of the garden?
A. 60 ft
B. [tex]$60 \sqrt{3} ft$[/tex]
C. 120 ft
D. [tex]$120 \sqrt{3} ft$[/tex]
Answer (1)
The shorter distance across the rhombus is twice the length of the shorter leg of the 3 0 ∘ − 6 0 ∘ − 9 0 ∘ triangle.
Calculate the length of the shorter leg: x = 2 30 = 15 feet.
Determine the hypotenuse of the triangle, which is the side length of the rhombus: 2 x = 30 feet.
Calculate the perimeter of the rhombus: 4 × 30 = 120 feet. 120 f t
Explanation
Problem Analysis The garden is shaped like a rhombus, composed of four identical 3 0 ∘ − 6 0 ∘ − 9 0 ∘ triangles. The shorter distance across the rhombus is 30 feet. We need to find the perimeter of the garden.
Find the shorter leg The shorter distance across the rhombus is formed by the shorter legs of two 3 0 ∘ − 6 0 ∘ − 9 0 ∘ triangles. Let x be the length of the shorter leg. Then, 2 x = 30 feet.
Calculate x Solving for x , we get x = 2 30 = 15 feet.
Determine the hypotenuse In a 3 0 ∘ − 6 0 ∘ − 9 0 ∘ triangle, the sides are in the ratio 1 : 3 : 2 . The shorter leg is x , the longer leg is x 3 , and the hypotenuse is 2 x . The hypotenuse of the triangle forms the side of the rhombus.
Calculate the side length of the rhombus The hypotenuse is 2 x = 2 ( 15 ) = 30 feet. This is the side length of the rhombus.
Calculate the perimeter The perimeter of the rhombus is 4 times the side length, so the perimeter is 4 × 30 = 120 feet.
Examples
Rhombus shapes are common in tile patterns and garden designs. Knowing how to calculate the perimeter of a rhombus, especially when it's formed from specific triangles, helps in estimating the amount of material needed to border a tiled area or fence a garden. For example, if you're creating a diamond-shaped flower bed using 3 0 ∘ − 6 0 ∘ − 9 0 ∘ triangles and know the shorter diagonal, you can quickly determine how much edging you'll need to enclose it.
