Problem 17 - 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. $$ a=5, c=10 $$

Step-by-Step Solution

Answer

The missing length of side 'b' is approximately 8.66 units.

Step 1: Understand the Pythagorean theorem

The Pythagorean theorem is \(a^2 + b^2 = c^2\) where 'a' and 'b' are the two sides and 'c' is the hypotenuse. For this exercise, 'a' is given as 5 and 'c' is given as 10.

Step 2: Substitute the given values

Substitute 'a' as 5 and 'c' as 10 into the Pythagorean theorem equation, reformulating it to \(5^2 + b^2 = 10^2\).

Step 3: Solve for b

After substituting, you get \(25 + b^2 = 100\). To solve for 'b', subtract 25 from each side of the equation. This modifies the equation to \(b^2 = 100 - 25\), which simplifies further to \(b^2 = 75\). To solve for 'b', take the square root of each side, resulting in \(b = \sqrt{75}\). Simplifying the square root of 75 gives the approximate value for 'b'.

Key Concepts

Right Triangle

A right triangle is a type of triangle that features a special angle: the right angle, which is exactly 90 degrees. This angle is typically marked with a small square inside the triangle's corner.
Because of this precise angle, right triangles are unique in their properties.

• They always have one right angle and two acute angles (each less than 90 degrees).
• The sides opposite these angles are called 'legs', while the side opposite the right angle is called the 'hypotenuse'.

The right triangle is significant in math because it allows the use of the Pythagorean Theorem—a crucial formula in geometry. This theorem is especially useful when needing to find an unknown length of one side, provided the lengths of the other two are known.

Hypotenuse

The hypotenuse is the longest side of a right triangle and is opposite the right angle. In the context of the Pythagorean Theorem, it is denoted as 'c' in the formula: \(a^2 + b^2 = c^2 \).
The hypotenuse plays a central role in calculating the missing lengths of the other two sides of the triangle. Here's what makes it important:

• It gives a direct relationship with the other two sides, allowing the use of the equation in calculations.
• Understanding its properties helps in validating the use of the Pythagorean Theorem correctly. For example, if calculated incorrectly, the hypotenuse should never be shorter than any of the other two sides.

In our exercise, we identified the hypotenuse as 10, which was then used to find the missing side 'b' of the triangle using the theorem.

Solving Equations

Solving equations is a fundamental skill in mathematics, used to find unknown values. When applying the Pythagorean Theorem, you'll often solve equations as seen in the given problem.
Let's review the steps to clearly solve the equation:

• First, substitute the known values into the Pythagorean equation \(a^2 + b^2 = c^2 \). In this scenario, we plugged in the values to become \(5^2 + b^2 = 10^2 \).
• Next, perform arithmetic operations to simplify the equation: \(25 + b^2 = 100\).
• Isolate the variable by subtracting 25 from each side to focus on 'b': \(b^2 = 75\).
• Find the square root of the resulting value, as this reveals the precise length of side 'b'. Hence, \(b = \sqrt{75} \) gives us the approximate length after simplification.

By following these straightforward steps to solve equations, students can confidently tackle similar problems involving right triangles and beyond.