Problem 28 - Prealgebra

Question

Writing in Math Suppose you have two triangles. Triangle \(A\) is similar to triangle \(B,\) and the measures of the sides of triangle \(A\) are less than the measures of the sides of triangle \(B\). The scale factor is \(0.25 .\) Which is the original triangle? Explain.

Step-by-Step Solution

Answer

Triangle B is the original triangle because Triangle A's sides are smaller by a scale factor of 0.25.

Step 1: Understand Similar Triangles

When two triangles are similar, their corresponding angles are equal and their corresponding sides are proportional. The ratio of the lengths of the corresponding sides is the scale factor.

Step 2: Given Information

You are given two triangles, Triangle A and Triangle B, which are similar. It is known that the side lengths of Triangle A are less than the side lengths of Triangle B, and the scale factor from Triangle A to Triangle B is 0.25.

Step 3: Interpret the Scale Factor

A scale factor of 0.25 indicates that each side length of Triangle A is 0.25 times (or one quarter of) the corresponding side length of Triangle B. This means Triangle A is smaller than Triangle B.

Step 4: Determine the Original Triangle

Since the side lengths of Triangle A are less and the scale factor given is less than 1, this means Triangle A is the reduced version of Triangle B. Therefore, Triangle B must be the original larger triangle.

Key Concepts

Scale Factor

The scale factor is a term used to compare the size of similar geometric figures. In the context of similar triangles, it represents the ratio between corresponding side lengths. In this exercise, we learn that Triangle A and Triangle B are similar, with Triangle A being smaller. The scale factor from Triangle A to Triangle B is 0.25, or one-quarter. This means that each side of Triangle A is 25% of the length of the corresponding side of Triangle B. When the scale factor is less than 1, it indicates that one triangle is a scaled-down version of the other. Here, Triangle A is smaller, meaning that by multiplying its side lengths by 4 (since 1 divided by 0.25 equals 4), you would get the side lengths of Triangle B. This tells us that Triangle B is the original, larger triangle.

Corresponding Angles

In similar triangles, one of the most important properties is that corresponding angles are always equal. This means that if two triangles are similar, every angle in one triangle has the exact same measure as the corresponding angle in the other triangle. In terms of appearance, this assures that both triangles maintain the same shape, differing only in size.

• This equality in angles does not change with the scale factor; whether one triangle is smaller or larger, the angles remain identical.
• Knowing that corresponding angles are the same gives you confidence in identifying similar triangles based on angle measurements alone.
• Therefore, even if the side lengths are scaled down or up, the angles remain constant, contributing to the similarity of the triangles.

Proportional Sides

When we say the sides are proportional in similar triangles, we mean that the lengths of corresponding sides are related by the scale factor. For each pair of corresponding sides, the ratio of their lengths will be equal to the scale factor.
This aspect of proportionality is pivotal in determining the relationship between the two triangles:

• For instance, if the side of Triangle A is 3 and the scale factor is 0.25, then the corresponding side of Triangle B would be 3 divided by 0.25, which is 12.
• Understanding this proportion is crucial when given a scale factor, as it allows you to derive unknown side lengths easily by setting up and solving proportional equations.

This keeps the triangles similar and shows that even though dimensions differ, the basic requirements for similarity are met, maintaining the same shape and angle integrity.