Problem 15 - 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. $$ b=3, c=7 $$

Step-by-Step Solution

Answer

The length of side 'a' is \(2\sqrt{10}\) units.

Step 1: Understanding the Pythagorean theorem

The Pythagorean theorem is normally expressed as \(a^2 + b^2 = c^2\). Here, \(a\) and \(b\) are the sides of the right triangle while \(c\) is the hypotenuse.

Step 2: Substituting the known values

Substitute the given lengths of side \(b=3\) and hypotenuse \(c=7\) into the Pythagorean theorem. It will look like this: \(a^2 + 3^2 = 7^2\).

Step 3: Solving for 'a'

After the substitution, we can solve for 'a'. First, simplify the equation to \(a^2 + 9 = 49\), then subtract 9 from both sides to solve for \(a^2\). The equation becomes \(a^2 = 40\). Then find the square root of 40 to solve for 'a', which will give \(a = \sqrt{40}\). Finally, simplify the square root to get the most simplified form \(a = 2\sqrt{10}\).

Key Concepts

Right Triangle

In geometry, one of the most fundamental shapes is the right triangle. A right triangle is a type of triangle that has one angle measuring exactly 90 degrees, also known as a right angle. The two other angles are acute, meaning they are each less than 90 degrees.

In right triangles, the side opposite the right angle is called the hypotenuse, and it is always the longest side. The other two sides, which form the right angle, are referred to as 'legs.' They are usually labeled as 'a' and 'b.' These two legs have important roles: one could be considered the "base," and the other, the "height."

Right triangles are immensely useful in computational geometry, physics, and engineering due to their simple properties and the important relationships expressed through the Pythagorean theorem.

Hypotenuse

The hypotenuse is the longest side of a right triangle. It is directly opposite the right angle itself, making it a key piece when using the Pythagorean theorem.

The significance of the hypotenuse arises from its involvement in the square relationship expressed in the theorem. This relationship, given by the equation \(c^2 = a^2 + b^2\), ties the squares of the lengths of all three sides of the triangle together. Knowing the length of the hypotenuse is crucial, whether you are solving for a missing leg or when making measurements in real-world applications like construction or navigation.

Always remember, in any right triangle, no matter how large or small, the hypotenuse holds this unique position and relationship, helping tie together the rest of the triangle's geometry.

Solving for a Variable

When faced with a right triangle problem, such as the exercise above, finding a missing side length requires solving for a variable using the Pythagorean theorem. This involves replacing known values into the equation and then isolating the unknown variable.

Let's break down the steps:

• Start by writing the Pythagorean theorem: \(a^2 + b^2 = c^2\).
• Substitute known values into the equation. For instance, if \(b=3\) and \(c=7\), then \(a^2 + 3^2 = 7^2\).
• Simplify the expression step-by-step. First solve \(3^2 = 9\) and \(7^2 = 49\), resulting in \(a^2 + 9 = 49\).
• Next, isolate \(a^2\) by subtracting 9 from both sides, resulting in \(a^2 = 40\).
• Finally, find the value of 'a' by calculating the square root of 40, resulting in \(a = \sqrt{40} = 2\sqrt{10}\).

This process of solving for one side can apply to any of the triangle's sides when two are known, illustrating the powerful utility of the Pythagorean theorem.