The two diagonals of a rhombus are 16 inches and 12 inches long. What is the perimeter of the rhombus? Round to the nearest inch, if necessary.
a. 28 in b. 40 in c. 56 in d. 66 in
divide into four right triangles
6, 4 are legs
so 3,4,5 triangles, hypotenuse is rhombus side = 10
10*4 = 40
To find the perimeter of a rhombus, you need to know the length of one side since all four sides of a rhombus are equal in length.
In this case, we are given the lengths of the diagonals. Let's call the length of the longer diagonal D1 and the length of the shorter diagonal D2.
The diagonals of a rhombus bisect each other at right angles, creating four right triangles. Since the diagonals are perpendicular bisectors, they split the rhombus into four congruent right triangles.
We can find the length of one side of the rhombus using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's label the unknown side of one right triangle as x. Since the diagonals bisect each other at right angles, each of the right triangles in the rhombus is a 30-60-90 triangle. In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2.
In our case, we can determine that D1 is the hypotenuse, so it is equal to 16 inches. D2 is the shorter side, so it is equal to 12 inches. We want to find the length of one side, which we will label as x.
Using the ratio 1 : √3 : 2, we can set up the following equation:
12/1 = x/√3
Cross-multiplying, we get:
12√3 = x
Simplifying, we find:
x ≈ 20.7846
Now, we have the length of one side of the rhombus. Since all sides of a rhombus are equal, the perimeter can be found by multiplying the length of one side by 4.
Perimeter = 4 * x ≈ 4 * 20.7846 ≈ 83.1383
Rounding to the nearest inch, the perimeter of the rhombus is approximately 83 inches.
None of the given answer choices are exactly 83 inches, so none of them are correct.
divide into four right triangles
6, 8 are legs
so 3,4,5 triangles, hypotenuse is rhombus side = 10
10*4 = 40