Q45. - Algebra 1

Question

MULTIPLE REPRESENTATIONS In this problem, you will explore how changing the lengths of the sides of a shape by a factor changes the perimeter of that shape.

1. GEOMETRIC Draw a square ABCD. Measure and label the sides. Draw a second square MNPQ with sides twice as long as ABCD. Draw a third square FGHJ with sides half as long as ABCD.
2. TABULAR Complete the table below using the appropriate measures.
   
   
   
3. VERBAL Make a conjecture about the change in the perimeter of a square if the side length is increased or decreased by a factor.

Step-by-Step Solution

Answer

a. The required figures are shown below.

b. The complete table is shown below:

c. The perimeter of a square is directly proportional to the side length. The perimeter increases as side length increases and the perimeter decreases as side length decreases.

Part a. Step 1. Given Information.

Side of MNPQ $=$ Twice the side of ABCD

Side of FGHJ $=$ Half the side of ABCD

Part a. Step 2. Draw the square ABCD.

Let the side of ABCD be $x$ meters.

Part a. Step 3. Draw the square MNPQ.

The side of MNPQ will be $2x$ meters.

Part a. Step 4. Draw the square FGHJ.

The side of FGHJ will be $\frac{x}{2}$ meters.

Thus, the above figures are the required squares.

Part b. Step 1. Given Information.

Side of MNPQ $=$ Twice the side of ABCD

Side of FGHJ $=$ Half the side of ABCD

Part b. Step 2. Side length and perimeter of square ABCD.

Let the side of ABCD be $x$ meters.

Perimeter of the square will be:

$P_{ABCD}=4x$

Part b. Step 3. Side length and perimeter of square MNPQ.

The side of MNPQ will be $2x$ meters.

Perimeter of the square will be:

$$\begin{gathered} P_{MNPQ}=4\left(2x\right) \\ {P}_{MNPQ}=8x \end{gathered}$$

Part b. Step 4. Side length and perimeter of square FGHJ.

The side of FGHJ will be $\frac{x}{2}$ meters.

Perimeter of the square will be:

$$\begin{gathered} P_{FGHJ}=4\left(\frac{x}{2}\right) \\ {P}_{FGHJ}=2x \end{gathered}$$

Thus, the required complete table is shown below:

Part c. Step 1. Given Information.

Side of MNPQ $=$ Twice the side of ABCD

Side of FGHJ width="18" style="max-width: none; vertical-align: -3px;" $=$ Half the side of ABCD

Part c. Step 2. Table of value.

The table of side length and perimeter is shown below:

Part c. Step 3. Relation between side length and perimeter.

It can be seen that perimeter is directly proportional to the side length of the square. That is, if the side length is doubled, the perimeter gets doubled and if the side length is halved, the perimeter is halved.

 

Thus, the perimeter of a square is directly proportional to the side length. The perimeter increases as side length increases and the perimeter decreases as side length decreases.