Question

essential question

1. exploration: perimeter

how can you solve an equation that has variables on both sides?
in each pair of diagrams below the polygons have the same perimeter. write an equation that can be used to find the value of x, and then find the perimeter of each figure.
a)
b)
c)
reflect
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Explanation:

Step1: Find perimeter of first polygon in a)

Perimeter of trapezoid is $(x + 1)+5 + 5+(x + 7)=2x+18$.

Step2: Find perimeter of second polygon in a)

Perimeter of triangle is $(x + 5)+(x + 5)+(x + 2)=3x + 12$.

Step3: Set perimeters equal and solve for x

Set $2x+18=3x + 12$. Subtract $2x$ from both sides: $18=x + 12$. Then subtract 12 from both sides, $x = 6$.

Step4: Find perimeter

Substitute $x = 6$ into either perimeter formula. Using the trapezoid's perimeter formula: $2(6)+18=30$.

Step5: Find perimeter of first polygon in b)

Perimeter of rectangle is $2((x + 5)+(2x+3))=2(3x + 8)=6x+16$.

Step6: Find perimeter of second polygon in b)

Perimeter of rectangle is $2((x + 1)+(3x - 1))=2(4x)=8x$.

Step7: Set perimeters equal and solve for x

Set $6x+16=8x$. Subtract $6x$ from both sides: $16 = 2x$. Then $x = 8$.

Step8: Find perimeter

Substitute $x = 8$ into either perimeter formula. Using the first rectangle's perimeter formula: $6(8)+16=64$.

Step9: Find perimeter of first polygon in c)

Perimeter of pentagon is $5(2x+5)=10x + 25$.

Step10: Find perimeter of second polygon in c)

Perimeter of hexagon is $6(3x+4)=18x+24$.

Step11: Set perimeters equal and solve for x

Set $10x + 25=18x+24$. Subtract $10x$ from both sides: $25=8x+24$. Then subtract 24 from both sides: $1 = 8x$, so $x=\frac{1}{8}$.

Step12: Find perimeter

Substitute $x=\frac{1}{8}$ into the pentagon's perimeter formula: $10(\frac{1}{8})+25=\frac{10}{8}+25=\frac{5}{4}+25=\frac{5 + 100}{4}=\frac{105}{4}=26.25$.

Answer:

a) $x = 6$, Perimeter = 30
b) $x = 8$, Perimeter = 64
c) $x=\frac{1}{8}$, Perimeter = 26.25