Problem 21 - Algebra 1: Concepts and Skills

Question

USING THE PYTHAGOREAN THEOREM Find the missing length of the right triangle if \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. $$ b=3, c=10 $$

Step-by-Step Solution

Answer

The length of the missing side a of the right triangle is \(\sqrt{91}\).

Step 1: Write down the Pythagorean theorem

In a right-angled triangle, the Pythagorean theorem can be written as \(c^{2}=a^{2}+b^{2}\). Here c is the length of the hypotenuse, a and b are the lengths of the other two sides.

Step 2: Substitute the known values into the formula

We know the lengths of side b=3 and the hypotenuse (c)=10. So, the formula becomes \(10^{2} = a^{2} + 3^{2}\).

Step 3: Solve for the unknown side a

First, compute the squares: \(100 = a^{2} + 9\). Rearrange this to find \(a^{2}\): \(a^{2} = 100 - 9 = 91\). Then, find the square root to determine the value of a: \(a = \sqrt{91}\). In this example, we will leave the answer in the radical form to be most accurate. However, one could approximate the result with a decimal if needed.

Key Concepts

Right Triangle

A right triangle, sometimes called a right-angled triangle, is a type of triangle that has one angle measuring exactly 90 degrees. This is known as the right angle. The right triangle is fundamental in geometry and is often encountered in various mathematical calculations.
In a right triangle:

• One angle is a perfect right angle.
• The other two angles must sum up to 90 degrees.
• It has three sides: the two shorter sides are called legs, and the longest side is called the hypotenuse.

The right triangle is significant for understanding the Pythagorean Theorem, which relates to the lengths of its sides. This theorem is particularly useful in identifying missing side lengths, especially when the triangle is in a coordinate system or any real-world context. Right triangles appear in many scenarios such as in carpentry, architecture, and even in video game graphics because knowing just two sides allows us to find the third side.

Hypotenuse

The hypotenuse is the longest side of a right triangle and is always directly opposite the right angle. It is a core component in the Pythagorean Theorem, which allows us to link the hypotenuse to the other two sides of a right triangle. The main properties of the hypotenuse include:

• The largest side in any right triangle.
• Always located across from the 90-degree angle.
• Critical in calculating distance and various real-world measurements.

When you use the Pythagorean Theorem, the hypotenuse is represented by the letter \(c\), and the theorem is written as \(c^{2} = a^{2} + b^{2}\). This relationship helps find a missing side when two of the triangle's sides' lengths are known.
For example, in the given problem, the hypotenuse length is already known to be 10, which becomes the foundation in applying the theorem to find the unknown side.

Legs of a Triangle

The legs of a right triangle are the two sides that form the right angle. These are typically denoted as \(a\) and \(b\) in the Pythagorean Theorem. Understanding the role of the legs in a right triangle is crucial as they help establish the foundation for exploring relationships within the triangle. Here's what you need to know:

• Each leg meets at the right angle.
• Their combined square lengths equal the square of the hypotenuse's length.
• Often the sides with known lengths when calculating the hypotenuse.

In the solved exercise, one leg length is known \(b = 3\), and the hypotenuse is \(c = 10\). Substituting these into the Pythagorean equation \(c^{2} = a^{2} + b^{2}\) allows the easy solving for the unknown leg \(a\). First, you calculate \(10^{2} = 100\), then subtract \(3^{2} = 9\) from 100 to find \(a^{2} = 91\). The side \(a\) itself is then found by taking the square root \(a = \sqrt{91}\), providing a precise radical answer that maintains mathematical accuracy.