Problem 80 - Elementary Algebra

Question

The perimeter of triangle \(A B C\) is 60 units and \(b=20\) units. If \(A B C \sim R S T\) and \(s=10\) units, then find the perimeter of triangle RST.

Step-by-Step Solution

Answer

The perimeter of triangle RST is 30 units.

Step 1: Understand the Problem

We have two similar triangles, \( \triangle ABC \sim \triangle RST \). Given the perimeter of \( \triangle ABC \) as 60 units and one side, \( b = 20 \) units. For \( \triangle RST \), we know one side \( s = 10 \) units. We need to find the perimeter of \( \triangle RST \).

Step 2: Establish the Ratio of Similarity

Since the triangles are similar, the corresponding sides of both triangles are proportional. The ratio between corresponding sides is given by \( \frac{b}{s} = \frac{20}{10} = 2 \). This indicates every corresponding side in \( \triangle ABC \) is twice as long as in \( \triangle RST \).

Step 3: Calculate the Ratio of Perimeters

The ratio of perimeters of similar triangles is equal to the ratio of their corresponding sides. Hence, the perimeter ratio is \( \frac{P_{ABC}}{P_{RST}} = 2 \), where \( P_{ABC} \) is the perimeter of triangle \( \triangle ABC \).

Step 4: Solve for the Perimeter of Triangle RST

Let \( P_{RST} \) be the perimeter of \( \triangle RST \). From the ratio, we have \( 60 = 2 \times P_{RST} \). Solving this equation for \( P_{RST} \), we get \( P_{RST} = \frac{60}{2} = 30 \) units.

Key Concepts

Perimeter of Triangle

The perimeter of a triangle is the total length of its three sides. It's similar to the way you measure a boundary, just focused on triangles.
To find it, simply add up the lengths of all sides. For a triangle like \( ABC \) with sides \( a, b, \) and \( c \), the perimeter \( P \) is expressed as \( P = a + b + c \).
In this specific exercise, the given perimeter of triangle \( ABC \) is 60 units, which is the sum of all its sides. When you know the perimeter, you can use it to compare with other similar triangles, just as we did with triangle \( RST \). This helps determine unknown quantities like missing side lengths or other perimeters.

Ratio of Similarity

In geometry, the ratio of similarity is like a link between two similar figures. For triangles, if they are similar, their corresponding sides are in proportion. This means one triangle is a scaled-up or scaled-down version of the other.
To know the ratio of similarity, compare the lengths of any pair of corresponding sides. In this problem, we calculated it as 2 by using the corresponding sides \( b \) and \( s \) from triangles \( ABC \) and \( RST \), respectively.

• Given: \( b = 20 \) units, \( s = 10 \) units.
• Thus, the ratio is \( \frac{b}{s} = \frac{20}{10} = 2 \).

This ratio tells us that each side of \( \triangle ABC \) is twice as long as each corresponding side of \( \triangle RST \).
It's like knowing the scale factor between a model and its actual size: simple yet revealing!

Corresponding Sides

Corresponding sides occur in similar triangles where pairs of sides are matched based on their relative position and length. Imagine two triangles \( ABC \) and \( RST \) that share similar shapes but differ in size.
Here, sides are considered corresponding if they face the same angle in similar triangles. So, side \( AB \) in \( \triangle ABC \) corresponds to side \( RS \) in \( \triangle RST \), and this applies similarly to other sides.
When solving problems like finding the perimeter or verifying similarity, it's key to identify and use these corresponding sides effectively.
By understanding this relation, calculations and comparisons between triangles become straightforward and clear.