Problem 32 - Prealgebra

Question

If \(c\) is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. $$a=?, b=\sqrt{123}=, c=22$$

Step-by-Step Solution

Answer

The missing measure for \(a\) is 19.

Step 1: Understand the Problem

You are given a right triangle with one leg, \(b\), expressed as \(b = \sqrt{123}\), and the hypotenuse \(c = 22\). Your goal is to find the measure of the other leg \(a\).

Step 2: Use the Pythagorean Theorem

For any right triangle, according to the Pythagorean Theorem, the relationship between the legs and the hypotenuse is \(a^2 + b^2 = c^2\). Here, plug in the given values: \(a^2 + (\sqrt{123})^2 = 22^2\).

Step 3: Calculate \(b^2\) and \(c^2\)

Evaluate \(b^2\) and \(c^2\). Since \(b = \sqrt{123}\), \(b^2 = 123\). For the hypotenuse, \(c = 22\), so \(c^2 = 22^2 = 484\).

Step 4: Solve for \(a^2\)

Substitute \(b^2 = 123\) and \(c^2 = 484\) into the equation \(a^2 + 123 = 484\). Simplify to find \(a^2 = 484 - 123 = 361\).

Step 5: Find \(a\)

To find \(a\), take the square root of \(a^2\). Therefore, \(a = \sqrt{361}\). Calculate \(a = 19\).

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle measuring exactly 90 degrees. This right angle is the reason it's called a "right triangle." Because of this distinct angle, the right triangle has unique properties, especially related to its sides and angles.
The two sides on either side of the right angle are called "legs," and the side opposite the right angle is known as the "hypotenuse." This hypotenuse is always the longest side in a right triangle since it spans directly across from the largest angle.
In a right triangle, you can explore fascinating relationships like the Pythagorean Theorem, which connects the lengths of all sides.

Hypotenuse

The hypotenuse is the longest side of a right triangle and it lies opposite the right angle. It is crucial in various calculations because it helps establish the relationship between the triangle's sides.
Understanding the hypotenuse is key to using the Pythagorean Theorem effectively. In mathematical terms, if you know the lengths of the two legs of a right triangle, you can find the hypotenuse using the formula: \[ c = \sqrt{a^2 + b^2} \] In our example, the hypotenuse is given as 22. This information allows us to find the other unknown side using the triangle's inherent properties.

Square Root

The square root is a mathematical way of determining which number, when multiplied by itself, will result in the given number. It's essential for solving problems involving the Pythagorean Theorem.
The square root symbol is \(\sqrt{}\). For example, \(\sqrt{123} \) represents the number that, when squared, equals 123.
Finding square roots is also a necessary step in determining side lengths when working with a right triangle. When solving for a side, such as when computing the missing leg, taking the square root is the final step in solving for its length.

Leg of a Triangle

In a right triangle, the two shorter sides are known as the "legs." These legs are crucial for calculations involving the triangle and provide the basis for the Pythagorean Theorem.
Each leg is perpendicular to the other, forming the right angle in the triangle. In our example, we have been given one leg as \(b = \sqrt{123}\). The job is to find the missing leg \(a\) with the help of the theorem.
To recall, the Pythagorean Theorem states: \[ a^2 + b^2 = c^2 \] By substituting the known values, you solve for the unknown, which in this case is \(a\) using simple algebraic manipulations.