Final answer:
To prove that quadrilateral JKLM is a kite, we calculated the lengths of the sides using the distance formula and found that there are two pairs of adjacent sides that are equal. Since JK equals MJ and KL equals LM, the definition of a kite is satisfied.
Step-by-step explanation:
To show that quadrilateral JKLM is a kite, we need to prove that there are two pairs of adjacent sides that are equal in length. Let's consider the given vertices J(0,0), K(0,a), L(4a,4a) and M(a,0). We can use the distance formula to find the lengths of the sides.
- For JK, the distance between J(0,0) and K(0,a) is |a-0|=a.
- For KL, the distance is √((4a-0)² + (4a-a)²) = √(16a² + 9a²) = √(25a²) = 5a.
- For LM, the distance between L(4a,4a) and M(a,0) is √((a-4a)² + (0-4a)²) = √(9a² + 16a²) = √(25a²) = 5a.
- For MJ, the distance between M(a,0) and J(0,0) is |a-0|=a.
Since JK = MJ = a and KL = LM = 5a, we can see there are two pairs of adjacent sides that are of equal length. Sides JK and MJ form one pair, and sides KL and LM form the second pair.
Therefore, JKLM is a kite by definition.