A kite has diagonals of 16 inches and 23 inches as seen below. What is the perimeter of the kite?
To find the perimeter of the kite, we need to know the lengths of all four sides. However, only the lengths of the diagonals are given. Therefore, we need to use the properties of a kite to find the missing side lengths.
A kite is a quadrilateral with two pairs of equal-length adjacent sides. In addition, the diagonals of a kite are perpendicular and intersect at a right angle.
Given that the diagonals of the kite are 16 inches and 23 inches, we can use these diagonals to find the lengths of the sides.
Let's call one of the diagonals "d1" and the other diagonal "d2". Since the diagonals bisect each other at right angles, they divide the kite into four congruent right triangles.
Using the Pythagorean theorem, we can find the length of each side:
Side 1: The diagonal d1 is the hypotenuse of one of the right triangles. Let's call the other two sides of the triangle "a" and "b". Applying the Pythagorean theorem, we have:
a^2 + b^2 = d1^2
a^2 + b^2 = 16^2
a^2 + b^2 = 256
Side 2: Using the same logic, let's call the sides of the second right triangle "c" and "d":
c^2 + d^2 = d2^2
c^2 + d^2 = 23^2
c^2 + d^2 = 529
Since the diagonals bisect each other, sides a and c are equal, and sides b and d are equal. Therefore, we can write the equation as:
a^2 + a^2 = 256
2a^2 = 256
a^2 = 128
a = sqrt(128) = 8sqrt(2)
Similarly:
c^2 + c^2 = 529
2c^2 = 529
c^2 = 264.5
c = sqrt(264.5) = sqrt(4 * 66.125) = sqrt(4) * sqrt(66.125) = 2sqrt(66.125) ≈ 16.24
Now, we have the lengths of the sides as follows:
Side 1 = Side 3 = 8√2 inches
Side 2 = Side 4 = 16.24 inches
To find the perimeter, we can add up the lengths of all four sides:
Perimeter = Side 1 + Side 2 + Side 3 + Side 4
Perimeter = (8√2 + 16.24) + (8√2 + 16.24)
Perimeter = 16√2 + 32.48 + 16√2 + 32.48
Perimeter ≈ 32 + 64.96 + 32 + 64.96 (approximating √2 as 1.41)
Perimeter ≈ 193.92 inches
Therefore, the perimeter of the kite is approximately 193.92 inches.