Problem 24 - Prealgebra

Question

The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. $$a=\sqrt{21}, b=6, c=\sqrt{57}$$

Step-by-Step Solution

Answer

The triangle is a right triangle.

Step 1: Identify the Longest Side

First, identify the longest side of the triangle since it will be considered as the hypotenuse if the triangle is a right triangle. Comparing \( a = \sqrt{21} \), \( b = 6 \), and \( c = \sqrt{57} \), we find that \( c = \sqrt{57} \) is the longest side.

Step 2: Apply the Pythagorean Theorem

According to the Pythagorean theorem, for a triangle to be a right triangle, \( c^2 \) should equal \( a^2 + b^2 \). Let's calculate each: \( a^2 = (\sqrt{21})^2 = 21 \), \( b^2 = 6^2 = 36 \). Add them to get \( a^2 + b^2 = 21 + 36 = 57 \).

Step 3: Compare with the Longest Side

Calculate \( c^2 \) and compare it with \( a^2 + b^2 \). \( c^2 = (\sqrt{57})^2 = 57 \). Since \( c^2 = a^2 + b^2 \), the given triangle is a right triangle.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that applies specifically to right-angled triangles. It states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Expressed in a formula, it is written as:\[ c^2 = a^2 + b^2 \]where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.To determine if a triangle is a right triangle using the Pythagorean theorem, you need to:

• Calculate the square of the apparent longest side.
• Calculate the squares of the other two sides.
• Add the results of these two calculations.

If the sum matches the square of the longest side, the triangle is a right triangle. This simple yet powerful formula is foundational in many mathematical applications, including algebra and trigonometry.

longest side of a triangle

Before applying principles like the Pythagorean theorem, it is crucial to identify the longest side of a triangle. For a right triangle, this side is known as the hypotenuse. In general, it holds special importance because it tells us where the right angle is located. To determine the longest side:

• Compare the lengths of all sides.
• The side with the greatest length will be the hypotenuse if the triangle is a right triangle.

Understanding that the longest side is the hypotenuse can simplify problem-solving. It helps confirm that where you suspect the right angle is, both in geometry problems and real-world scenarios such as construction or when using measurements.

right-angled triangle properties

A right-angled triangle has distinct properties that differentiate it from other types of triangles. The most defining one is its right angle, which is exactly 90 degrees. Here are some key properties:

• The hypotenuse is opposite the right angle and is always the longest side.
• The other two angles are acute, meaning each measures less than 90 degrees.
• The Pythagorean theorem holds true, allowing for precise calculations of side lengths.

These properties are not just abstract; they have practical applications. Right triangles are used in various fields such as architecture, navigation, and more. When identifying or working with these triangles, remembering their characteristics ensures accurate results in calculations and designs.