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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

3. Tests for Parallelograms

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Exercise 23 Page 501

Recall the Parallelogram Opposite Sides Theorem.

x=4
y=3

Practice makes perfect

Let's find the values of both variables at one time.

Values of x and y

Notice that the sides with lengths 6y+ 12x and 2x+4y are opposite to each other. Moreover, the sides with lengths 21 and 3x+3y are also opposite to each other. Let's recall the Parallelogram Opposite Sides Theorem.

Parallelogram Opposite Sides Theorem
If opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Therefore, if we want the given quadrilateral to be a parallelogram, the opposite sides must be congruent. 6y+ 12x=2x+4y & (I) 21=3x+3y & (II) We will solve the system by using the Substitution Method. Let's start by isolating the x-variable in Equation (II).

6y+ 12x=2x+4y & (I) 21=3x+3y & (II)

▼

Solve for x

(II): .LHS /3.=.RHS /3.

6y+ 12x=2x+4y 7=x+y

(II): LHS-y=RHS-y

6y+ 12x=2x+4y 7-y=x

(II): Rearrange equation

6y+ 12x=2x+4y x=7-y

Now that x is isolated in Equation (II), we will substitute x=7-y in Equation (I), and solve for y.

6y+ 12x=2x+4y x=7-y

(I): x= 7-y

6y+ 12( 7-y)=2( 7-y)+4y x=7-y

▼

Solve for y

(I): Distribute 1/2

6y+ 72- y2=2( 7-y)+4y x=7-y

(I): Distribute 2

6y+ 72- y2=14-2y+4y x=7-y

(I): LHS * 2=RHS* 2

12y+7-y=28-4y+8y x=7-y

(I): Add and subtract terms

11y+7=28+4y x=7-y

(I): LHS-4y=RHS-4y

7y+7=28 x=7-y

(I): LHS-7=RHS-7

7y=21 x=7-y

(I): .LHS /7.=.RHS /7.

y=3 x=7-y

We found that y=3. Finally, we will find the value of x by substituting this value in Equation (II).

y=3 x=7-y

(II): y= 3

y=3 x=7- 3

(II): Subtract term

y=3 x=4

Subchapter links

Exercises p.499-503