Question

Suppose \(\triangle R S T \sim \triangle U V W\) and the scale factor of \(\triangle R S T\) to \(\triangle U V W\) is \(\frac{3}{2}\). Find the perimeter of \(\triangle U V W\) if the perimeter of \(\triangle R S T\) is 57 inches.

Short answer

The perimeter of \(\triangle UVW\) is 38 inches.

Step-by-step solution

Understand the Relationship Between Similar Triangles

Given that \(\triangle RST \sim \triangle UVW\), we know they are similar triangles, meaning corresponding sides have proportional lengths, and their ratios are equal to the scale factor.

Identify the Scale Factor Relationship

The scale factor from \(\triangle RST\) to \(\triangle UVW\) is \(\frac{3}{2}\). This means for every 3 units in \(\triangle RST\), there are 2 corresponding units in \(\triangle UVW\).

Apply the Scale Factor to the Perimeter

Since the scale factor describes the ratio of corresponding sides, it also applies to the perimeters. Thus, if the perimeter of \(\triangle RST\) is 57 inches, the perimeter of \(\triangle UVW\) will be scaled down by the factor \(\frac{2}{3}\) (the inverse of \(\frac{3}{2}\)).

Calculate the Perimeter of \(\triangle UVW\)

The formula to find the perimeter of \(\triangle UVW\) is \(\frac{2}{3} \times 57\). Performing the multiplication: \[\frac{2}{3} \times 57 = 38\]. Therefore, the perimeter of \(\triangle UVW\) is 38 inches.

Key concepts

Understanding Scale Factor in Triangles

Scale factor is a crucial concept when working with similar triangles. It describes how much one triangle has been resized to form another, while maintaining the same shape. This is often represented as a ratio or fraction. In the given exercise, the scale factor from \(\triangle RST\) to \(\triangle UVW\) is \(\frac{3}{2}\). This means that every length in \(\triangle RST\) corresponds to a length in \(\triangle UVW\) that is two-thirds of the original.

• A scale factor greater than 1 indicates an enlargement.
• A scale factor less than 1 indicates a reduction.

Understanding the scale factor helps in resizing figures proportionally, which is especially helpful in calculating dimensions such as side lengths or the perimeter of the similar triangle.

Calculating Triangle Perimeter with the Scale Factor

The perimeter of a triangle is simply the sum of all its sides. For similar triangles, the perimeters are directly proportional to their corresponding side lengths. This means that if you know the perimeter of one triangle, you can find the perimeter of a similar triangle using the scale factor.
In the original exercise, we found that if \(\triangle RST\) has a perimeter of 57 inches and a scale factor of \(\frac{3}{2}\), we can use the inverse of the scale factor, \(\frac{2}{3}\), to determine \(\triangle UVW\)'s perimeter. The calculation would look like this:
\[\frac{2}{3} \times 57 = 38\] inches.
Therefore, understanding how to manipulate the scale factor allows you to determine missing measures efficiently.

Proportional Relationships in Similar Triangles

Proportional relationships are the backbone of similar triangles. These relationships mean that the ratios of the lengths of corresponding sides are equal. The known scale factor \(\frac{3}{2}\) implies that for each unit of length in \(\triangle RST\), \(\triangle UVW\) has a proportional length.

• If a side of \(\triangle RST\) is 3 units, the corresponding side in \(\triangle UVW\) would be 2 units.
• This consistent ratio applies not just to the side lengths, but also to other measurements like altitude and medians, if these components are involved.

Mastering proportional relationships allows students to solve various geometric problems, as they can extend these concepts to different scenarios involving similar shapes.