Big Ideas Math Geometry, 2014
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Big Ideas Math Geometry, 2014 View details
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Exercise 8 Page 411

Recall the classification of quadrilaterals. You can begin by finding the lengths of its sides using the Distance Formula.

Rhombus

Practice makes perfect

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.