Chapter Review
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Begin by drawing parallelogram JKLM on a coordinate plane.
Square, Rectangle, and Rhombus
We have been given a parallelogram JKLM with vertices J(5,8), K(9,6), L(7,2), and M(3,4). Let's first plot the vertices and draw the parallelogram on a coordinate plane.
Remember that a rhombus is a parallelogram with four congruent sides. Therefore, we will use the Distance Formula to find the side lengths of JKLM.
| Side | Distance Formula | Simplify |
|---|---|---|
| Length of JK: ( 5,8), ( 9,6) | sqrt(( 9- 5)^2+( 6- 8)^2) | sqrt(20) |
| Length of KL: (9,6), (7,2) | sqrt((7-9)^2+(2-6)^2) | sqrt(20) |
| Length of LM: (7,2), ( 3,4) | sqrt(( 3-7)^2+( 4-2)^2) | sqrt(20) |
| Length of MJ: (3,4), (5,8) | sqrt((5-3)^2+(8-4)^2) | sqrt(20) |
The lengths of the sides are equal, so the four sides are congruent. Therefore, the parallelogram is a rhombus.
To see whether the parallelogram is a rectangle, we will recall a theorem that tells us that if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Let's look at the lengths of the diagonals of the parallelogram.
| Diagonal | Distance Formula | Simplify |
|---|---|---|
| Length of JL: ( 5,8), (7,2) | sqrt((7- 5)^2+(2- 8)^2) | sqrt(40) |
| Length of MK: (3,4), (9,6) | sqrt((9-4)^2+(6-4)^2) | sqrt(40) |
The lengths of the diagonals are equal, so the diagonals are congruent. Therefore, the parallelogram is a rectangle.
By the Square Corollary, we know that a quadrilateral is a square if and only if it is a rhombus and a rectangle. We showed that JKLM is both a rhombus and a rectangle. Therefore, we can conclude that it is also a square.