Problem 17 - Algebra 1

Question

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 length of the missing side 'a' in the right triangle is \( \sqrt{175} \).

Step 1: Understand the Problem

This problem is a classic case of applying the Pythagorean theorem, which can be defined as \(a^2 + b^2 = c^2\) where, a and b are the lengths of the legs of a right triangle and c is the hypotenuse. Here, b and c values are given, b=9 and c=16, we need to find the length of a.

Step 2: Rearrange the Pythagorean theorem

We can write the Pythagorean theorem focusing on 'a'. Doing so, we get \(a = \sqrt{c^2 - b^2}\). The value of 'a' can be found by subtracting the square of 'b' from the square of 'c' and taking the square root of the result.

Step 3: Plugging values

Inserting the given b and c values into the equation, we get \(a = \sqrt{16^2 - 9^2} = \sqrt{256 - 81} = \sqrt{175}\). Therefore, the length of 'a' is \(\sqrt{175}\).

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one of its angles measuring exactly 90 degrees. This particular angle makes it unique as it allows us to utilize the Pythagorean theorem to compute one of its side lengths if the other two are known. Right triangles are often seen in geometry and trigonometry, serving as basic building blocks for more complex shapes and concepts.
In a right triangle:

• The two sides that form the right angle are known as the 'legs'.
• The longest side, opposite the right angle, is referred to as the 'hypotenuse'.

When solving problems involving right triangles, identify these sides first. This approach makes applying the Pythagorean theorem straightforward, and correctly identifying the right triangle is crucial in executing geometric analyses.

Hypotenuse

The hypotenuse is the longest side in a right triangle, lying directly opposite the right angle. It is a key element in the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (\(c^2\)) is equal to the sum of the squares of the other two sides (legs), represented as \(a^2 + b^2\).
Some characteristics of the hypotenuse include:

• It is always longer than either of the triangle's legs.
• In the Pythagorean theorem, it's denoted by 'c'.

Understanding the hypotenuse is crucial because it allows us to calculate the length of the other sides when applied within the theorem's formula.

Square Root

A square root is a value that, when multiplied by itself, gives the original number. In the context of the Pythagorean theorem, calculating the side lengths of a right triangle often requires finding the square root of the sum or difference of squared numbers.
Here's a bit more about square roots:

• The square root of a number \(x\) is represented as \(\sqrt{x}\).
• Finding square roots is essential for solving for a side in a right triangle when using the Pythagorean theorem.

When solving triangle problems, ensuring precise calculation of square roots is critical. This helps find the exact lengths of sides, especially when the length isn't a perfect square, as seen in our solution with \(\sqrt{175}\).