Chapter Test
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Recall the classification of quadrilaterals. You can begin by finding the lengths of its sides using the Distance Formula.
Rhombus
Let's plot the given points on a coordinate plane and graph the quadrilateral.
We will determine the most precise name for our quadrilateral. To do so, let's review the classification of quadrilaterals.
| Quadrilateral | Definition |
|---|---|
| Parallelogram | Both pairs of opposite sides are parallel |
| Rhombus | Parallelogram with four congruent sides |
| Rectangle | Parallelogram with four right angles |
| Square | Parallelogram with four congruent sides and four right angles |
| Trapezoid | Quadrilateral with exactly one pair of parallel sides |
| Isosceles Trapezoid | Trapezoid with legs that are congruent |
| Kite | Quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent |
Now, let's find the side lengths of our quadrilateral using the Distance Formula.
| Side | Distance Formula | Simplified |
|---|---|---|
| Length of PQ: ( 5,1), ( 9,6) | sqrt(( 9- 5)^2+( 6- 1)^2) | sqrt(41) |
| Length of QR: ( 9,6), (5, 11) | sqrt((5- 9)^2+(11- 6)^2) | sqrt(41) |
| Length of RS: (5, 11), (1, 6) | sqrt((1-5)^2+(6-11)^2) | sqrt(41) |
| Length of SP : (1, 6), ( 5,1) | sqrt(( 5-1)^2+( 1-6)^2) | sqrt(41) |
Our quadrilateral has four congruent sides. Therefore, it is either a square or a rhombus. To check this, we can find the slopes of the sides using the Slope Formula.
| Side | Slope Formula | Simplified |
|---|---|---|
| Slope of PQ: ( 5,1), ( 9,6) | 6- 1/9- 5 | 5/4 |
| Slope of QR: ( 9,6), (5, 11) | 11- 6/5- 9 | - 5/4 |
| Slope of RS: (5, 11), (1,6) | 6-11/1-5 | 5/4 |
| Slope of SP: (1, 6), ( 5,1) | 1-6/5-1 | - 5/4 |
We can see that the slopes of the opposite sides of our quadrilateral are equal. Therefore, both pairs of opposite sides are parallel and the quadrilateral is a parallelogram. We can also tell that the consecutive sides are not perpendicular, as their slopes are not opposite reciprocals.
5/4*( - 5/4 ) ≠ -1
Therefore, our quadrilateral is a rhombus.