Question

Scale Drawings A window measures 8 feet by 3 feet. Make a scale drawing of the window if \(\frac{1}{4}\) inch represents 1 foot. (Lesson 9-2)

Short answer

Draw the window as a 2-inch by \(\frac{3}{4}\)-inch rectangle.

Step-by-step solution

Determine the Scale Factor

A scale of \(\frac{1}{4}\) inch represents 1 foot. This means each foot in real life corresponds to \(\frac{1}{4}\) inch in the drawing.

Calculate the Scaled Length for Width

The actual width of the window is 8 feet. Using the scale factor, convert this to a scaled length: \(8 \text{ feet} \times \frac{1}{4} \text{ inch/foot} = 2 \text{ inches}\) in the drawing.

Calculate the Scaled Length for Height

The actual height of the window is 3 feet. Using the scale factor, convert this to a scaled length: \(3 \text{ feet} \times \frac{1}{4} \text{ inch/foot} = \frac{3}{4} \text{ inch}\) in the drawing.

Draw the Scaled Rectangle

Using the scaled dimensions calculated, draw a rectangle that is 2 inches wide and \(\frac{3}{4}\) inch tall to represent the window in the scale drawing.

Key concepts

Scale Factor

Understanding the concept of a scale factor is crucial when working with scale drawings. A scale factor is a number that scales or multiplies some quantity. In the context of scale drawings, it refers to the ratio between any two corresponding lengths in the real world and the drawing. For instance, if the scale is \( \frac{1}{4} \) inch representing 1 foot, this implies that the drawing measures are \( \frac{1}{4} \) of the actual size. This scale factor is used to transform real-life dimensions into their scaled representation.
To utilize a scale factor, multiply each real-life dimension by the scale factor:

• Real width: 8 feet becomes \( 8 \times \frac{1}{4} = 2 \) inches in the drawing.
• Real height: 3 feet becomes \( 3 \times \frac{1}{4} = \frac{3}{4} \) inch in the drawing.

Conversions like these help create accurate scale drawings, preserving the original shape's proportions on a smaller scale.

Geometry

Geometry plays an essential role when creating and understanding scale drawings. It involves understanding shapes, sizes, and the properties of space. When dealing with tasks like scaling windows, geometry ensures that the shape and proportions remain consistent even after resizing.
A window is typically a rectangle, having two dimensions: width and height. In the exercise, the window measures 8 feet by 3 feet. Even when scaled down, keeping the angles and ratios consistent is important to accurately reflect the original shape. For example:

• Both width and height are divided by the same scale factor, maintaining the window's rectangular shape and proportions.
• By drawing a rectangle with scaled dimensions (2 inches by \( \frac{3}{4} \) inch), you ensure it visually represents the original window in a smaller form.

Geometry principles ensure that the essence of the original shape is preserved in the scaled version.

Proportional Reasoning

Proportional reasoning is understanding the constant multiplicative relationship between numbers. When dealing with scale drawings, it's essential to maintain proportion between real-life dimensions and their scaled representations. This means if one dimension scales down, the others must follow by the same scale factor to keep the proportions equal.
Applying proportional reasoning allows you to:

• Consistently apply the scale factor to all dimensions, ensuring accurate representation in drawings.
• Understand that if one dimension reduces, all associated measurements reduce proportionately.
• Visualize real-world objects in a simplified form while keeping different dimensions in harmony.

In our exercise, by scaling both width and height by \( \frac{1}{4} \), we draw a window that is proportionally accurate to the original, preserving the relationship between the height and width ratios. This fundamental principle of proportional reasoning ensures that despite changes in size, the essence of the object remains the same.