Question

Suppose \(\triangle A B C \sim \triangle D E F\) and the scale factor of \(\triangle A B C\) to \(\triangle D E F\) is \(\frac{5}{3}\). Find the perimeter of \(\triangle D E F\) if the perimeter of \(\triangle A B C\) is 25 meters.

Short answer

The perimeter of \(\triangle DEF\) is 15 meters.

Step-by-step solution

Understanding Similar Triangles

Two triangles are similar if they have the same shape but not necessarily the same size. In this problem, \(\triangle ABC \sim \triangle DEF\), meaning the corresponding sides of the triangles are proportional, and the ratio of their perimeters is the same as the scale factor.

Identifying the Scale Factor

The problem states that the scale factor from \(\triangle ABC\) to \(\triangle DEF\) is \(\frac{5}{3}\). This means every dimension of \(\triangle ABC\) is \(\frac{5}{3}\) times longer than the corresponding dimension of \(\triangle DEF\).

Applying the Scale Factor to Perimeter

The perimeter of one triangle is proportional to the perimeter of the similar triangle by the scale factor. Given the perimeter of \(\triangle ABC\) is 25 meters, we can find the perimeter of \(\triangle DEF\) by dividing the perimeter of \(\triangle ABC\) by the scale factor \(\frac{5}{3}\).

Calculating the Perimeter of Triangle DEF

Divide the perimeter of \(\triangle ABC\) by the scale factor to find the perimeter of \(\triangle DEF\): \[ P_{\triangle DEF} = \frac{P_{\triangle ABC}}{\frac{5}{3}} = \frac{25}{\frac{5}{3}} = 25 \times \frac{3}{5} = 15 \text{ meters} \]

Key concepts

Scale Factor

When we talk about the scale factor in geometry, we're referring to the ratio that compares the sizes of two similar figures. In the context of triangles, like \(\triangle ABC \sim \triangle DEF\), the scale factor tells us how much one triangle is enlarged or reduced relative to the other. If the scale factor from \(\triangle ABC\) to \(\triangle DEF\) is \(\frac{5}{3}\), this means that each side of \(\triangle ABC\) is \(\frac{5}{3}\) times the length of the corresponding side of \(\triangle DEF\). The scale factor also applies to every linear measure of the triangle, not just the sides. This includes perimeters, altitudes, and medians, among others.
To use the scale factor effectively, remember it is:

• A multiplier for increasing or decreasing dimensions.
• A ratio that stays constant for all corresponding measurements.

A larger scale factor means the figure is enlarged, while a smaller scale factor indicates a reduction.

Perimeter

The perimeter of a triangle is simply the sum of the lengths of its three sides. When dealing with similar triangles, the perimeters are also proportional to one another because of the constant scale factor. This proportionality allows us to calculate the unknown perimeter if we know the perimeter of the other triangle and the scale factor.

How to Calculate Using Scale Factor

If you know the perimeter of \(\triangle ABC\) is 25 meters and the scale factor from \(\triangle ABC\) to \(\triangle DEF\) is \(\frac{5}{3}\), you can find the perimeter of \(\triangle DEF\) by adjusting \(P_{\triangle ABC}\) using the inverse of the scale factor:
\[ P_{\triangle DEF} = \frac{P_{\triangle ABC}}{\frac{5}{3}} = 25 \times \frac{3}{5} = 15 \text{ meters} \]
This adjustment shrinks dimensions proportionally to match the smaller triangle, reflecting how the perimeters relate.

Proportionality in Geometry

Proportionality is a key concept in understanding similar triangles. When triangles are similar, their corresponding sides, angles, and line segments are proportional. This means there’s a constant ratio—like the scale factor—that applies across all these dimensions.

Understanding Proportions

With proportionality:

• Corresponding sides of similar triangles maintain a constant ratio.
• Corresponding angles remain the same in measurement, preserving shape.
• Any line segment within the triangle, such as an altitude, will be proportional between the similar triangles.

This proportionality holds for dimensions beyond just length, affecting areas and volumes when considering 3D shapes as well.
To leverage proportionality effectively, it's crucial to establish the scale factor first, as it will underpin all other proportional comparisons and calculations.