Introduction to Proof: Reasoning in Geometry

Geometric Induction

The following figures are called kites. In a kite, each pair of sides at the end of only one diagonal is equal. For example, in the following kites, each pair of sides at the end of vertical diagonals is equal.  Recall that matching hatch marks on individual shapes are congruent.  In the figures below, the pairs of sides that are congruent are marked.

four kites labeled P with diagonals 2 and 7, Q with diagonals 4 and 5, S with diagonals 6 and 14 and R with diagonals 7 and 10

In these kites, the lengths of diagonals are given. Using a computer program, we found the following measures for their areas:

Observation 1

Area of P = 7
Area of Q = 10
Area of R = 35
Area of S = 42

Observation 2

Let's try to develop a pattern using these numbers in terms of the lengths of their diagonals.

Area of P = 7 = 7 × 1
Area of Q = 10 = 5 × 2
Area of R = 35 = 7 × 5
Area of S = 42 = 14 × 3

We can represent the areas of the kites as a product where one of the factors is one of the diagonals.

Observation 3

Let's try to use the multiplication problems above to find the other diagonal. We can write the second factor in each product as 1/2 of the other diagonal.

Area of P = 7 = 7 × 1 = 7 × one-half × 2
Area of Q = 10 = 5 × 2 = 5 × one-half × 4
Area of R = 35 = 7 × 5 = 7 × one-half × 10
Area of S = 42 = 14 × 3 = 14 × one-half × 6

Observation 4

The area of each of these kites is one-half of the product of its diagonals.

Conjecture

To find the area of a kite, we simply multiply the diagonals and then find one-half of the result. That is, if the diagonals of a kite measure a and b, then its area, A, is calculated by A = one-halfab.