jmc

Question

Kite ABCD (a quadrilateral with two pairs of adjacent equal sides) has coordinates A (0,7), B (1,0), C (12,-2), and D (7,8). What is the area of ABCD, given that the area of a kite is equal to half the product of its diagonals? FIGURE

Accepted Answer

As the problem suggests, we need to compute the lengths of the diagonals

\overline{A C}

and

\overline{B D}

. By the distance formula,

A C B D = (12 - 0)^{2} + (- 2 - 7)^{2} = 1 2^{2} + 9^{2} = 15 = (7 - 1)^{2} + (8 - 0)^{2} = 6^{2} + 8^{2} = 10

Thus, the answer is

\frac{1}{2} \cdot 10 \cdot 15 = 75

.

As an extra challenge, can you figure out why the area of a kite equals half the product of the lengths of its diagonals?