Problem 16 - 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. $$ b=9, c=16 $$

Step-by-Step Solution

Answer

The missing length of the right triangle, \(a\), is \(5\sqrt{7}\).

Step 1: Identification of Given Variables

Identify the given variables in the problem. Here \(b = 9\) and \(c = 16\).

Step 2: Rearrange the Pythagorean formula

Rearrange the Pythagorean theorem formula to isolate \(a\). This gives us the formula \(a^2 = c^2 - b^2\).

Step 3: Substituting values into the formula

Substitute \(b = 9\) and \(c = 16\) into the formula. This gives us \(a^2 = 16^2 - 9^2\).

Step 4: Simplify the equation

Simplify the equation to solve for \(a^2\). We get \(a^2 = 256 - 81\).

Step 5: Solve for \(a^2\)

Subtract 81 from 256 to get \(a^2 = 175\).

Step 6: Solve for \(a\)

Taking the square root of both sides to give \(a = \sqrt{175}\), which simplifies to \(a = 5\sqrt{7}\).

Key Concepts

Understanding Right Triangles

A right triangle is a type of triangle that includes one angle measuring exactly 90 degrees, often called a right angle. This type of triangle is unique because it forms the basis of the Pythagorean Theorem.

• In a right triangle, the side opposite the right angle is known as the hypotenuse, which is always the longest side.
• The other two sides are referred to as the "legs" of the triangle.

Understanding the components of a right triangle is crucial for solving problems related to them, especially when using mathematical formulas like the Pythagorean Theorem. If you know two side lengths, you can determine the third using this powerful theorem. Remember that in any right triangle, the Pythagorean Theorem is applied only to these specific triangles where one angle is exactly 90 degrees.

The Role of the Hypotenuse

The hypotenuse has a special role in the context of right triangles. Often labeled as "c" in mathematical problems, the hypotenuse is integral to employing the Pythagorean Theorem. Here’s why the hypotenuse matters:

• It is always opposite the right angle.
• In the Pythagorean Theorem, it is the side used to determine the relationship with the legs, labeled as "a" and "b."
• Mathematically, the square of the hypotenuse (\(c^2\)) equals the sum of the squares of the other two sides (\(a^2 + b^2\)).

When calculating the missing length of a triangle using this theorem, knowing the hypotenuse can help determine the unknown leg. This is why it's important to recognize which side is the hypotenuse to correctly apply the theorem.

Applying the Square Root Function

The square root function is a mathematical operation that is often used in conjunction with the Pythagorean Theorem. Let’s break down what happens when you take the square root:

• If you square a number, you have multiplied it by itself. The square root reverses this process.
• In our problem, once we calculated \(a^2 = 175\), taking the square root allows us to find the actual length of side "a."
• The square root of 175 simplifies to \(5\sqrt{7}\) using factorization techniques.

The square root function is essential for working with the Pythagorean Theorem because it transforms squared numbers back into their original lengths, completing the process of solving for missing sides. Remember, only non-negative results apply when calculating side lengths in right triangles.