Problem 79 - Elementary Algebra

Question

A 6-8-10 right triangle \(A B C\) is similar to a triangle RST with perimeter 72 units. Find the length of each leg of triangle RST.

Step-by-Step Solution

Answer

Triangle RST has legs 18, 24, and 30 units long.

Step 1: Identify Corresponding Parts

Triangle ABC is a right triangle with sides 6, 8, and 10. Triangle RST is similar to triangle ABC. In similar triangles, corresponding sides are proportional. Therefore, we set up a proportion using the sides of triangle ABC to find the sides of triangle RST.

Step 2: Calculate the Scale Factor

Since triangle RST is similar to triangle ABC, there exists a scale factor, which we can determine by using the perimeter. The perimeter of triangle ABC is \(6 + 8 + 10 = 24\). The scale factor \(k\) is the ratio of the perimeters of the triangles: \(k = \frac{72}{24} = 3\).

Step 3: Calculate Side Lengths of Triangle RST

Using the scale factor \(k = 3\), multiply each side of triangle ABC to find the corresponding side lengths of triangle RST. The sides of triangle RST are:\(RS = 6 \times 3 = 18\),\(ST = 8 \times 3 = 24\),\(TR = 10 \times 3 = 30\).

Key Concepts

Right Triangle

In geometry, a right triangle is a triangle that has one angle measuring exactly 90 degrees, known as a right angle. This special angle creates two legs that are perpendicular to each other and a hypotenuse, which is the longest side of the triangle opposite the right angle. The Pythagorean Theorem is a fundamental property of right triangles, stating that the square of the hypotenuse is equal to the sum of the squares of the other two sides. For example, in a right triangle ABC with sides 6, 8, and 10, the relationship can be shown as:

• The legs: 6 and 8
• The hypotenuse: 10

According to the Pythagorean Theorem: \(6^2 + 8^2 = 10^2 \), which verifies that it is indeed a right triangle.
In this context, recognizing right triangles helps in understanding the geometric relationships and proportional reasoning used in the exercise.

Scale Factor

A scale factor is a number that scales, or multiplies, some quantity. In the context of similar triangles, it is the ratio that describes how much one triangle is enlarged or reduced to form a similar triangle. If two triangles are similar, their sides are proportional, and a single scale factor can be used to transition from one to the other.

• In the provided exercise, triangle ABC with a perimeter of 24 is expanded to form triangle RST with a perimeter of 72.
• The scale factor \(k\) is calculated by dividing the perimeters: \(k = \frac{72}{24} = 3\).

This means that each side of triangle ABC is multiplied by 3 to get the corresponding side of triangle RST. Understanding scale factors is crucial for solving problems involving similar figures efficiently. They allow us to easily calculate unknown side lengths once the scale factor is known.

Proportional Sides

When two triangles are similar, their respective sides are proportional. This concept means the corresponding sides have the same ratio or are "scaled" by the same factor. For example, if triangle ABC has sides of lengths 6, 8, and 10, and triangle RST is similar, each side of RST would be a multiple or proportionate of the sides in ABC.

• The side 6 in ABC corresponds to 18 in RST (as 6 times the scale factor 3 equals 18).
• Similarly, side 8 becomes 24, and side 10 becomes 30.

This proportionality is fundamental in determining unknown lengths in geometry. It allows us to express and solve equations to find missed information about similar shapes or figures. Recognizing these relationships helps simplify the process of solving for the unknown sides of a similar figure.