Question

If rectangle ABCD were enlarged by a factor of 3 (so that its sides became 3 times as long as they were previously), what would be the ratio of the new area to the old area? (A) 1:3 (B) 1:9 (C) 3:1 (D) 9:1

Short answer

The ratio of the new area to the old area when the rectangle is enlarged by a factor of 3 is 9:1.

Step-by-step solution

Understand the original and new rectangle dimensions

Let's suppose original rectangle ABCD has length 'l' and width 'w'. By enlarging by a factor of 3, the new dimensions of the rectangle will be 3l and 3w.

Calculate the area of the original and new rectangles

First, we need to find the area of both rectangles. For the original rectangle ABCD (with dimensions l and w): Area = l × w For the new rectangle (with dimensions 3l and 3w): New area = (3l) × (3w) = 9lw

Find the ratio of the new area to the old area

Now that we have both the original area (lw) and the new area (9lw), we can find the ratio between them. Ratio = New area : Old area = 9lw : lw As 'lw' is a common term in both, we can divide both sides of the ratio by 'lw' to simplify it. Ratio = 9 : 1 Therefore, the answer is (D) 9:1.

Key concepts

Understanding Area Calculation

When talking about area calculation, it's all about determining the size of a two-dimensional surface within the boundary of a shape. For a rectangle, the area is calculated by multiplying its length by its width. This is expressed using the formula: \[ \text{Area} = \text{length} \times \text{width} \] Consider rectangle ABCD with length \( l \) and width \( w \). The area of this rectangle is simply \( lw \). Calculating area is crucial when comparing the size of similar shapes, particularly in transformations.

Understanding the Scale Factor

A scale factor is a number used as a multiplier to increase or decrease the size of a shape. When you enlarge or shrink a geometric figure, the dimensions change by this factor. In our example, the scale factor is 3.

• If the length of the rectangle is \( l \), then the new length will be \( 3l \).
• If the width is \( w \), then it becomes \( 3w \).

Notice that when you apply the scale factor to both the length and width, the area must change accordingly. This change is represented by squaring the scale factor when calculating area.

Understanding Rectangular Dimensions

Rectangular dimensions refer to the aligned pairs of length and width that define a rectangle. For instance, rectangle ABCD has these dimensions: length \( l \) and width \( w \). In a geometric transformation where you enlarge a rectangle:

• The new length becomes \( 3l \).
• The new width becomes \( 3w \).

This enlargement directly influences the shape's area. The reduction or enlargement strongly affects use in practical settings like construction or crafting where spatial accuracy is necessary.