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McGraw Hill Glencoe Geometry, 2012

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McGraw Hill Glencoe Geometry, 2012 View details

Study Guide and Review

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Exercise 40 Page 452

Begin by finding the slopes of the sides.

Type of Quadrilateral: Rhombus
Explanation: See solution.

Practice makes perfect

Let's plot the given points on a coordinate plane and graph the quadrilateral.

To list all the names that apply for our parallelogram, let's recall the definitions of rhombus, rectangle and square.

Quadrilateral	Definition
Rhombus	Parallelogram with four congruent sides.
Rectangle	Parallelogram with four right angles.
Square	Parallelogram with four congruent sides and four right angles.

Now, let's find the slopes of the sides using the Slope Formula.

Slope Formula: y_2-y_1/x_2-x_1
Side	Endpoints	Substitute	Simplify
QR	Q( - 2,4) and R( 5,6)	6- 4/5-( - 2)	2/7
RS	R( 5,6) and S( 12,4)	4- 6/12- 5	- 2/7
ST	S( 12,4) and T( 5,2)	2- 4/5- 12	2/7
TQ	T( 5,2) and Q( - 2, 4)	4- 2/- 2- 5	- 2/7

We can tell that the consecutive side are not perpendicular, as their slopes are not opposite reciprocals. 2/7 ( - 2/7 ) ≠ -1 Therefore, our parallelogram is neither a rectangle nor a square. To check if it is a rhombus, we can find the lengths of its sides using the Distance Formula.

Distance Formula: sqrt((x_2-x_1)^2+(y_2-y_1)^2)
Side	Endpoints	Substitute	Simplify
QR	Q( - 2,4) and R( 5,6)	sqrt(( 5-( - 2))^2+( 6- 4)^2)	sqrt(53)
RS	R( 5,6) and S( 12,4)	sqrt(( 12- 5)^2+( 4- 6)^2)	sqrt(53)
ST	S( 12,4) and T( 5,2)	sqrt(( 5- 12)^2+( 2- 4)^2)	sqrt(53)
TQ	T( 5,2) and Q( - 2, 4)	sqrt(( - 2- 5)^2 + ( 4- 2)^2)	sqrt(53)

Our parallelogram has four congruent sides. Therefore, it is a rhombus.