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# 4.2: Variables on Both Sides

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Algebra I, Chapter 3, Lesson 4.

## Problem 1 – A Square and a Rectangle Have Different Perimeters.

A square has sides of length \begin{align*}x\end{align*}. A rectangle has one side that is twice as long and another that is \begin{align*}3\ units\end{align*} longer than the sides of the square. Do these expressions reflect the description in the picture to the right?

• Write an algebraic expression for the perimeter of the square to the right.
• Write an algebraic expression for the perimeter of the rectangle to the right.
• If the rectangle has a perimeter that is \begin{align*}10\ units\end{align*} longer than the perimeter of the square, which of the following equations are true?

a. \begin{align*}4x + 10 = 2(x + 3) + 2(2x)\end{align*}

b. \begin{align*}4x - 10 = 2(x + 3) + 2(2x)\end{align*}

c. \begin{align*}4x = (x + 3) + 2x + 10\end{align*}

d. none of these

• What value of \begin{align*}x\end{align*} will make the equation true?
• Check your answer using the App4Math application by pressing APPS and selecting App4Math. If your entered answer is correct, the calculator will display true.

Note: \begin{align*}x, y, z\end{align*}, etc. can be entered using the alpha keys.

Use o for the equals sign.

## Problem 2 – An Equilateral Triangle and a Square have Different Perimeters.

An equilateral triangle has sides of length \begin{align*}x\end{align*}. A square has sides that are \begin{align*}1\end{align*} more than twice that length. The perimeter of the square is \begin{align*}19\ centimeters\end{align*} more than that of the triangle.

• How long are the sides of each polygon?
• Write an algebraic expression for the perimeter of the square.
• Write an algebraic expression for the perimeter of the triangle.
• Write an equation that shows the relationship if the perimeters of the square and triangle.
• Solve this equation and state the length of each side of the square.

## Problem 3 – A Regular Hexagon and a Regular Octagon

A regular hexagon has sides of length \begin{align*}x\end{align*}. A regular octagon has sides that are half as long. The perimeter of the hexagon is \begin{align*}20\ inches\end{align*} longer than that of the octagon.

• If each side of the hexagon is of length \begin{align*}2x\end{align*}, what is the length of each side of the octagon?
• Write an algebraic expression for the perimeter of the hexagon.
• Write an algebraic expression for the perimeter of the octagon.
• Write an equation shows the perimeter of the hexagon and octagon, then find the length of the sides of the hexagon.

## Problem 4 – An Equilateral Triangle and a Rectangle

To the right is figure comprised of an equilateral triangle and a rectangle. The perimeter of the rectangle is \begin{align*}9\ centimeters\end{align*} more than the perimeter of the triangle.

• Find the length, \begin{align*}x\end{align*}, of each side of the triangle.

## Problem 5 – Regular Decagon and 15-gon

The side lengths of the regular decagon and \begin{align*}15-\end{align*}gon to the right are equal.

• Find the difference in their perimeters.

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