Problem 18 - 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. $$ a=14, c=21 $$

Step-by-Step Solution

Answer

The length of the missing side \(b\) of the right triangle is \(b = \sqrt{21^2 - 14^2}\)

Step 1: Understand the Given Information

The given information tells us that the length of side \(a\) of the right triangle is 14, and the length of the hypotenuse \(c\) is 21.

Step 2: Use the Pythagorean theorem to write down the Equation

Start by writing down the Pythagorean theorem as \(a^2 + b^2 = c^2\). Then insert the known values, which gives us \(14^2 + b^2 = 21^2\).

Step 3: Solve for \(b\)

Rearrange the updated equation from Step 2 to solve for \(b\). This results in \(b^2 = 21^2 - 14^2\). Now, compute the length of \(b\) by taking the square root of \(b^2\), which should give us \(b = \sqrt{21^2 - 14^2}\).\n

Key Concepts

Right Triangle

In geometry, a **right triangle** is a type of triangle that has one angle measuring exactly 90 degrees. This makes it unique compared to other triangles. The side opposite the right angle is known as the hypotenuse, while the other two sides are referred to as the legs. These triangles have special properties, and one of the most significant is their compatibility with the Pythagorean Theorem, which helps us find missing side lengths.
The right triangle is a fundamental shape in mathematics and is used in various practical applications like construction, design, and navigation. Understanding its properties allows us to solve many real-world problems involving it. The Pythagorean Theorem only works for these specific types of triangles, making them particularly important in geometric studies.

Hypotenuse

The **hypotenuse** is the longest side of a right triangle and is always found opposite the right angle. This side is crucial when using the Pythagorean Theorem to solve problems. In our example problem, the hypotenuse is given as 21 units.
Finding the hypotenuse or using it to find other sides is a common problem in geometry:

• For a given hypotenuse, we can determine the other side lengths using the Pythagorean Theorem: \(a^2 + b^2 = c^2\).
• If you know both legs, calculating the hypotenuse involves rearranging the equation to \(c = \sqrt{a^2 + b^2}\).

Understanding the hypotenuse's length and its function within the triangle is essential in applying the Pythagorean Theorem effectively.

Leg Length

In a right triangle, the **leg lengths** are the two shorter sides that meet at the right angle. In calculations, these legs are critical because understanding one helps find the other, especially when given the hypotenuse length. In our example, one leg is given as 14 units, and we're tasked with finding the missing leg.
The use of the Pythagorean Theorem simplifies this calculation:

• Insert the known leg and hypotenuse into the theorem \(\text{leg}_1^2 + \text{leg}_2^2 = \text{hypotenuse}^2\).
• Rearrange to solve for the unknown leg \(\text{leg}_2 = \sqrt{\text{hypotenuse}^2 - \text{leg}_1^2}\).

This gives us an efficient way to complete the triangle's side measurements, ensuring all components are correctly balanced as per their geometric properties.