Problem 21 - Prealgebra

Question

The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. $$a=16, b=30, c=34$$

Step-by-Step Solution

Answer

The triangle with sides 16, 30, and 34 is a right triangle.

Step 1: Identify the Hypotenuse

For a right triangle, the longest side is the hypotenuse. Among the sides given, identify the hypotenuse. Here, compare the side lengths: - a = 16 - b = 30 - c = 34 Since 34 is the largest, c = 34 is the hypotenuse.

Step 2: Apply the Pythagorean Theorem

To check if the triangle is a right triangle, apply the Pythagorean theorem: \[ c^2 = a^2 + b^2 \] Substitute the known values: \[ 34^2 = 16^2 + 30^2 \] Calculate both sides.

Step 3: Calculate Each Side of the Equation

First, calculate the squares of each side:1. Calculate \( c^2 \): \[ 34^2 = 1156 \]2. Calculate \( a^2 + b^2 \): \[ 16^2 + 30^2 = 256 + 900 = 1156 \]

Step 4: Compare the Results

Compare the results from Step 3:- \( c^2 = 1156 \)- \( a^2 + b^2 = 1156 \)Since both sides of the equation are equal, this confirms that the triangle is a right triangle.

Key Concepts

Understanding the Pythagorean Theorem

The Pythagorean Theorem is one of the most fundamental concepts in geometry, especially when dealing with right triangles. It provides a simple relationship between the sides of a right triangle. This theorem states that for any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides, often called the legs. In mathematical terms, it is expressed as: \[ c^2 = a^2 + b^2 \] where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs. The Pythagorean Theorem allows you to quickly determine if a triangle with known side lengths is a right triangle. All you have to do is check if the equation holds true for the given side lengths.

Hypotenuse Identification

In a right triangle, correctly identifying the hypotenuse is crucial for using the Pythagorean Theorem effectively. The hypotenuse will always be the longest side of the triangle. When given the side lengths, a quick comparison will help you determine which one it is. For example, if you have sides \( a = 16 \), \( b = 30 \), and \( c = 34 \), it’s clear that \( c \), being 34, is the hypotenuse because it is the largest number. Remember, this straightforward comparison is the first step whenever you want to verify if a triangle is a right triangle using the Pythagorean Theorem. Identifying the hypotenuse accurately ensures correct calculations and validates the conditions set by the theorem.

Triangular Side Lengths

When tasked with determining if a triangle is right-angled, understanding how to work with triangular side lengths is vital. First, list all the side lengths provided. Then, as discussed, identify the hypotenuse. Once you have the correct sides categorized, apply the Pythagorean Theorem by plugging in the values.After substituting the side lengths into \( c^2 = a^2 + b^2 \), compute each side of the equation:

• Calculate \( c^2 \)
• Calculate \( a^2 + b^2 \)

Compare the two results. If they are equal, the triangle is a right triangle. With the example of \( a = 16 \), \( b = 30 \), and \( c = 34 \), after calculation, both sides equal 1156, confirming the triangle as right-angled.Understanding these processes allows you to analyze equations easily and verify triangular properties effectively.