If the length of the shorter diagonal is four, what is the length of the longer diagonal of this kite?

Question: If the length of the shorter diagonal is four, what is the length of the longer diagonal of this kite?Solution:We can find the longer diagonal by adding together the altitude of the top triangle and the altitude of the bottom triangle. To find these, use Pythagorean Theorem. We can use Pythagorean Theorem because one of the properties of a kite is that the two diagonals are perpendicular.The top triangle has two sides of length 3 [labeled in the picture], and a base of 4 [provided in the written directions]. To figure out the altitude, split this triangle into 2 right triangles. The two legs are x [the altitude] and 2 [half of the base 4], and the hypotenuse is 3:x2 + 22 = 32⇒ x2 + 4 = 9⇒ x2 = 5∴ x = √5We will do something similar for the bottom triangle. Consider one of the right triangles. It will have a hypotenuse of 7, one leg that we don't know, x [the altitude], and one leg 2 [half the shorter diagonal]. Set up the equation using the Pythagorean Theorem:x2 + 22 = 72⇒ x2 + 4 = 49⇒ x2 = 45∴ x = √45 = 3√5∴ The length of the longer diagonal of this kite = 3√5 + √5 = 4√5