Q48. - Algebra 1

Question

Determine whether this statement is true or false. If the length of a rectangle is doubled, then the area of the rectangle is doubled. Justify your answer.

Step-by-Step Solution

Answer

The given statement is true.

Step 1. State the concept and formula.

A conditional statement is false when the hypothesis is true but the conclusion is false.

The area of a rectangle is given as $A=lw$, where $A$ is the area, $l$ is the length and $w$ is the width of the rectangle.

Step 2. List the given data.

The given statement is “If the length of a rectangle is doubled, then the area of the rectangle is doubled.”

Then the hypothesis is “length of a rectangle is doubled” and the conclusion is “area of the rectangle is doubled”.

Step 3. Calculate the area.

Let the hypothesis be true. So, the length of a rectangle is doubled.

Then, $l_{new}=2l$.

Put $l_{new}=2l$ in $A=lw$ to get,

$\begin{array}{l}{A}_{new}=l_{new}w\\ =2lw\\ =2\left(lw\right)\\ =2A\end{array}$

So, $A_{new}=2A$.

This implies that the area of the new rectangle is double that of the area of the initial rectangle.

So, the conclusion is true.

Step 4. Check if the statement is true.

It has been obtained that if the hypothesis is true, then the conclusion is true. This implies that the given statement is true.