Exercise 32 Page 795

We will find the perimeter and area of the given kite one at a time. Let's do it!

Perimeter

The perimeter of any polygon is calculated by adding all its side lengths. Recall that a kite is a quadrilateral with exactly two pairs of consecutive congruent sides. With this information, we also know the side lengths that are not given in the diagram. Let's add them!

We can now calculate the perimeter of the kite. We want to express the perimeter in feet. To do so, and considering that $1$ meter is approximately $3.28$ feet, we will multiply the obtained perimeter by $3.28.$ The perimeter of the polygon to the nearest tenth is $67.2$ feet.

Area

The area of a kite is half the product of its diagonals. To find the diagonals, we will start by considering the right triangle formed in the top-right portion of the diagram.

Note that the legs of the triangle are congruent. Therefore, we have an isosceles right triangle — also known as a $45^{\circ }\text{-}45^{\circ }\text{-}90^{\circ }$ triangle. In this type of special triangle, the length of each leg is $\frac{1}{\sqrt{2}}$ times the length of the hypotenuse. Knowing that the length of the hypotenuse is $3\sqrt{2}\text{ m},$ we can find the length of the legs. Let's add the newly obtained information to the diagram. Also, notice that the larger diagonal bisects the shorter diagonal.

By the Segment Addition Postulate, the length of the shorter diagonal is $3+3=6$ meters.

Let's now find the length of the larger diagonal. To do so, we will pay close attention to the right triangle formed in the top-left portion of the diagram.

In this right triangle, the length of the hypotenuse is $6$ meters and the length of one of the legs is $3$ meters. To find the length of the other leg, we will substitute these values into the Pythagorean Theorem. The length of the longer leg of the right triangle is $3\sqrt{3}$ meters. Therefore, by the Segment Addition Postulate, the length of the larger diagonal is $3\sqrt{3}+3$ meters. Using the diagonal lengths, let's find the area of the kite. Finally, to express the area in square feet, we will multiply the obtained number by the conversion factor $3.28$ twice. Let's do it! The area of the polygon to the nearest tenth is $264.5{\text{ ft}}^2.$