Problem 74 - Elementary Algebra

Question

If the 3-4-5 right triangle \(A B C\) is similar to \(R S T\) with a scale factor of 5 , then find the perimeter of triangle \(R S T\).

Step-by-Step Solution

Answer

The perimeter of triangle \( RST \) is 60 units.

Step 1: Understand the Similarity

The triangles \( \triangle ABC \) and \( \triangle RST \) are similar, meaning their corresponding angles are equal and their corresponding sides are in the same ratio. We are given that the scale factor is 5.

Step 2: Identify the Sides of \( \triangle ABC \)

The sides of \( \triangle ABC \) are 3, 4, and 5, forming a right triangle. Let's denote these sides as \( AB = 3 \), \( BC = 4 \), and \( CA = 5 \).

Step 3: Apply the Scale Factor to Find \( \triangle RST \) Sides

To find the sides of \( \triangle RST \), multiply each side of \( \triangle ABC \) by the scale factor 5. Thus, the sides of \( \triangle RST \) are \( 3 \times 5 = 15 \), \( 4 \times 5 = 20 \), and \( 5 \times 5 = 25 \).

Step 4: Calculate the Perimeter of \( \triangle RST \)

Add the lengths of the sides of \( \triangle RST \) to find its perimeter: \( 15 + 20 + 25 = 60 \).

Key Concepts

Right Triangles

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This 90-degree angle is called a right angle. Right triangles are special because they follow the Pythagorean theorem. This theorem states that the square of the length of the hypotenuse (the longest side opposite the right angle) is equal to the sum of the squares of the other two sides. For example, in a 3-4-5 right triangle, like \( \triangle ABC \), the sides are in the ratio of 3:4:5, which is a classic example following the Pythagorean theorem: \( 3^2 + 4^2 = 5^2 \), or \( 9 + 16 = 25 \). This specific pattern helps us easily identify and solve problems involving right triangles, and it's important to recognize these relationships when working with similar triangles.

Scale Factor

The scale factor is a crucial concept when dealing with similar triangles. When two triangles are similar, their corresponding angles are the same, and the lengths of their corresponding sides are proportional. The scale factor is the number that tells us how much one triangle has been enlarged or reduced compared to another.

For instance, if a small triangle has sides measuring 3, 4, and 5, and the scale factor is 5, we multiply each side of the small triangle by 5. This means each side is scaled up by this factor, giving the sides of the larger triangle as 15, 20, and 25. This simple multiplication helps in expanding our triangle while maintaining its original shape because the angles remain the same, adhering to the rules of similar triangles. Understanding the scale factor can make solving geometry problems much simpler.

Perimeter Calculation

The perimeter of a triangle is the total distance around the triangle, calculated by adding the lengths of all its sides. To find the perimeter of a triangle once the side lengths are known, simply add them together.

In this exercise, after determining the side lengths of \( \triangle RST \) using the scale factor (15, 20, and 25), we find the perimeter by adding these values: \( 15 + 20 + 25 = 60 \). Calculating the perimeter is essential in understanding the size and boundary of a triangle, and knowing how to find it quickly is a useful skill in geometry. This calculation offers insights not just into the triangle's dimensions but also into its overall geometry in space.