Problem 5 - 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. $$a=5, c=13$$

Step-by-Step Solution

Answer

The missing length (b) of the right triangle is 12.

Step 1: Identify the Given Values

In the problem, we are given the values of a and c. Specifically, \(a = 5\) and \(c = 13\). We are required to find the length of b

Step 2: Apply Pythagoras' Theorem

The formula that we will use is derived from the Pythagorean theorem and is \(b = \sqrt{c^{2} - a^{2}}\). Substituting \(a = 5\) and \(c = 13\) we get \(b = \sqrt{(13^{2}) - (5^{2})}\).

Step 3: Calculate the Result

Carry out the operation in the brackets first: \(b = \sqrt{(169) - (25)} = \sqrt{144}\). Find the square root of 144 to find the length of b. So, \(b = 12\).

Key Concepts

Understanding Right Triangle

In geometry, a right triangle is a triangle that has one angle measuring 90 degrees. The sides of a right triangle include two legs and one hypotenuse. The two legs are the sides that form the right angle, while the hypotenuse is the longest side opposite the right angle. Understanding the right triangle is crucial when applying the Pythagorean theorem.

• The legs are typically labeled as 'a' and 'b'.
• The hypotenuse is labeled as 'c'.
• Right triangles are fundamental in various fields like architecture, engineering, and astronomy.

It is important to remember that the Pythagorean theorem only applies to right triangles, making them a unique case in triangle geometry. Identifying the side lengths allows us to use this theorem to find missing measures.

Identifying the Hypotenuse

The hypotenuse is a special side in right triangles. As the longest side, it is located directly opposite the right angle. This distinction is vital when using the Pythagorean theorem.

• The hypotenuse is always greater than either of the legs.
• It plays a crucial role in trigonometry and geometry, especially when calculating areas and perimeters.

In our exercise, we are given that the hypotenuse is 13 units long. When labeling a right triangle, it is customary to use 'c' to denote the hypotenuse. Recognizing and identifying this side correctly is the first step in solving many geometry problems.

Exploring Triangle Geometry

Triangle geometry involves understanding various types and properties of triangles, with the right triangle often highlighted for its unique traits. When considering the geometry of triangles:

• There are three types of triangles based on angles: right, acute, and obtuse.
• The sum of angles in any triangle is always 180 degrees.

The right triangle is particularly interesting due to the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, or legs. This fundamental principle lays the groundwork for solving numerous geometric problems, including those involving side lengths, such as the exercise you've completed. Understanding these principles enhances your ability to tackle more complex geometric challenges.