Problem 22 - Prealgebra

Question

The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. $$a=18, b=24, c=30$$

Step-by-Step Solution

Answer

The triangle is a right triangle.

Step 1: Identify the Hypotenuse

In a right triangle, the hypotenuse is the side opposite the right angle and is the longest side. Compare the lengths of the sides given: 18, 24, and 30. Identify 30 as the longest side, which makes it the hypotenuse.

Step 2: Apply the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Form the equation using the given lengths: \[ a^2 + b^2 = c^2 \] Substituting the values, we get: \[ 18^2 + 24^2 = 30^2 \]

Step 3: Calculate Each Side's Square

Calculate the square of each side: \( 18^2 = 324 \) \( 24^2 = 576 \) \( 30^2 = 900 \)

Step 4: Sum and Compare with Hypotenuse Square

Add the squares of the two shorter sides: \( 324 + 576 = 900 \) Compare this to the square of the hypotenuse: \( 900 = 900 \) Since the sum is equal to the hypotenuse's square, it confirms the triangle is a right triangle.

Key Concepts

Pythagorean Theorem in Right Triangles

The Pythagorean theorem is a fundamental principle when it comes to right triangles. It helps us establish the relationship between the sides of a triangle where one angle is 90 degrees. This theorem simplifies the identification of right triangles by stating a specific rule: the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Mathematically, it can be represented as:

• Given a triangle with sides labeled as \( a \), \( b \), and \( c \) (where \( c \) is the hypotenuse), the equation is \( a^2 + b^2 = c^2 \).

If this equation holds, then we have a right triangle. It is a handy shortcut to verify the nature of a triangle's angles without needing to measure them directly.

Understanding the Hypotenuse

The hypotenuse is the longest side in a right triangle, located opposite the right angle. This is a key aspect when applying the Pythagorean theorem. In any triangle, the hypotenuse will always be crucial in determining and applying right triangle rules.

Here’s how you can identify the hypotenuse:

• Look at the triangle's sides and identify the longest one. This is your hypotenuse.
• In our problem, comparing the sides \(18\), \(24\), and \(30\), we notice that \(30\) is the longest, hence is the hypotenuse.

Recognizing the hypotenuse correctly is essential because it forms the basis of applying the Pythagorean theorem to determine right triangles.

Triangle Sides Square Calculation

Calculating the square of each side of the triangle is a straightforward but essential step in using the Pythagorean theorem. Here’s how you can compute and use these squares:

• Record the sides of your triangle. In our scenario, they are \( a = 18 \), \( b = 24 \), and \( c = 30 \).
• Calculate each square: \( 18^2 = 324 \), \( 24^2 = 576 \), and \( 30^2 = 900 \).
• Add the squares of the two shorter sides: \( 324 + 576 = 900 \).
• Compare this sum with the square of the hypotenuse: If they match, as \( 900 = 900 \) does here, the triangle is indeed a right triangle.

This process of calculating and comparing offers a simple check that refines your understanding of these geometric principles, solidifying your grasp of the right triangle concept.