Problem 16 - Elementary and Intermediate Algebra

Question

The lengths of two sides of the right triangle \(A B C\) are given. Find the length of the missing side. \(a=5 \mathrm{in.}\) and \(b=12 \mathrm{in.}\) (RIGHT TRIANGLE CAN'T COPY)

Step-by-Step Solution

Answer

The missing side of the triangle is 13 inches.

Step 1: Understanding the Problem

We need to find the length of the missing side of a right triangle where two sides are given: one side is 5 inches and the other is 12 inches. Since this is a right triangle, we can apply the Pythagorean theorem.

Step 2: Identify Hypotenuse or Leg

In a right triangle, the hypotenuse is the longest side and is opposite the right angle. We compare the given sides: 5 inches and 12 inches, and note that we need to determine if the missing side is the hypotenuse or another leg of the triangle.

Step 3: Apply Pythagorean Theorem as Leg Calculation

Since both given sides are possible legs (as one side is shorter than the other), calculate the hypotenuse using the Pythagorean theorem: \[ c = \sqrt{a^2 + b^2} \] where \( a = 5 \) and \( b = 12 \).

Step 4: Square the Known Sides

Calculate the squares of the known sides:\[ 5^2 = 25 \] \[ 12^2 = 144 \]

Step 5: Sum of Squares

Add together the squares of the known sides:\[ 25 + 144 = 169 \]

Step 6: Calculate the Hypotenuse

Take the square root of the sum to find the hypotenuse:\[ c = \sqrt{169} = 13 \] So, the hypotenuse or missing side is 13 inches.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This right angle is a key part of the triangle's identity and separates it from other types of triangles.
When dealing with a right triangle, it's important to understand its properties:

• One angle measures 90 degrees, forming a perfect L-shape.
• The side opposite the right angle is always the longest, called the hypotenuse.
• The other two sides, known as legs, meet at the right angle.

Recognizing the right triangle's structure is crucial when applying the Pythagorean theorem. In a typical problem like finding a missing side, knowing one angle is key. The right angle allows us to use unique formulas that are not applicable to other triangles. Always check a question to confirm it involves a right triangle before using this specialized knowledge.

Hypotenuse

The hypotenuse is a special term used to describe the longest side of a right triangle. It is found directly opposite the right angle. This side holds great significance because it is involved in many important calculations.
When using the Pythagorean theorem to find the hypotenuse, you involve these steps:

• Identify the two legs of the triangle.
• Apply the formula: \[ c = \sqrt{a^2 + b^2} \], where "c" represents the hypotenuse.
• Square each leg, sum them, and then compute the square root to determine "c".

In our problem, given that the other two sides measure 5 inches and 12 inches, calculating the hypotenuse simplifies the process of understanding the triangle's dimensions. Knowing the hypotenuse allows for further geometric and trigonometric exploration of the triangle.

Leg Calculation

Calculating the length of a leg in a right triangle involves identifying it and applying the Pythagorean theorem when necessary.
If you are given the hypotenuse and one leg but need to find the other leg, you adjust the formula:

• Use \[ a = \sqrt{c^2 - b^2} \] for the missing leg, where "c" is the hypotenuse and "b" is the known leg.
• Substitute the known values into the formula and solve.
• This involves subtracting the square of the known leg from the square of the hypotenuse before taking the square root.

In the case where you have both legs and need to find the hypotenuse, as in our exercise, you simply apply:\[ c = \sqrt{5^2 + 12^2} \] which reveals the hypotenuse to be 13 inches. Recognizing when a side is a leg or the hypotenuse streamlines the calculation process, ensuring accurate solutions for right triangles.