The diagonals of a kite are 21 and 24 and the shorter diagonal bisects the longer. If the kite has exactly one right angle, what are the side lengths of the kite?
the diagonals of a kite are perpendicular
the angles at the ends of the longer diagonal are congruent
the right angle is at one end of the shorter diagonal, and is bisected by it
... this means there are two isosceles right triangles on the longer side of the shorter diagonal
... so two sides of the kite are 12√2
the other two sides are ... √(12^2 + 9^2)
since when does the shorter diagonal bisect the longer?
Steve ... since the problem says so
To find the side lengths of the kite, we can use the properties of kites.
First, let's denote the longer diagonal as d1 and the shorter diagonal as d2. We are given that d1 = 24 and d2 = 21.
We know that the shorter diagonal bisects the longer diagonal, so we can divide d1 by 2 to find the length of each half of d1:
Length of half of d1 = d1 / 2 = 24 / 2 = 12
Next, let's denote the lengths of the two equal sides of the kite as s. Since we have one right angle, the two equal sides adjacent to the right angle are equal in length. Therefore, we have two sides with length s.
Now, we can use the Pythagorean theorem to find the length of the remaining side of the kite.
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's consider one of the right triangles formed by the diagonals of the kite. The longer diagonal, d1, is the hypotenuse, and the two equal sides, s, are the other two sides.
Using the Pythagorean theorem, we can express this as:
d1^2 = s^2 + s^2
24^2 = 2s^2
576 = 2s^2
Dividing both sides by 2, we get:
288 = s^2
Taking the square root of both sides, we find:
s = √288
Now, we simplify:
s = √(144 x 2)
s = √144 x √2
s = 12√2
Therefore, the length of each side of the kite is 12√2, and the side lengths of the kite are 12√2, 12√2, 21, and 24.