Problem 3 - Algebra 1: Concepts and Skills

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. $$ a=7, b=24 $$

Step-by-Step Solution

Answer

So, the length of the hypotenuse \(c\) is 25.

Step 1: Write down the Pythagorean theorem

The Pythagorean theorem is given by: \[c^2 = a^2 + b^2\]

Step 2: Substitute given values

Substitute the given values into the equation: \[c^2 = 7^2 + 24^2\] This simplifies to: \[c^2 = 49 + 576\]

Step 3: Simplify the equation

Simplify the equation by first adding the numbers on the right side: \[c^2 = 625\]

Step 4: Solve for the length of the hypotenuse

In order to find the length of the hypotenuse \(c\), take the square root of both sides of the equation: \[c = \sqrt{625}\]

Key Concepts

Understanding Right Triangles

A right triangle is a triangle in which one of the angles is exactly 90 degrees. This angle is known as the right angle. Right triangles have a unique property that makes them very interesting for solving various problems involving geometry. The sides of a right triangle are typically referred to as two 'legs' and the 'hypotenuse'.

• The legs are the two sides that form the right angle.
• The hypotenuse is the longest side and is opposite the right angle.

Right triangles are not only fundamental in geometry but also in practical applications like construction and physics. Their simplicity and symmetry make them very easy to work with, and this is where the Pythagorean Theorem comes into play.

What is the Hypotenuse?

The hypotenuse is the longest side of a right triangle. Since it is opposite the right angle, it stretches across the triangle, forming a diagonal. In every right triangle, the hypotenuse plays a crucial role in calculations related to the triangle's shape and form.
When using the Pythagorean Theorem, \(c\) is often used to denote the length of the hypotenuse. For example, in a right triangle with legs of lengths \(a\) and \(b\), the theorem states that the sum of the squares of these two legs equals the square of the hypotenuse. Thus: \[c^2 = a^2 + b^2\] Finding the length of the hypotenuse can be applied in numerous situations, including navigation, construction, and various fields of engineering.

Explaining the Square Root

The square root is a mathematical operation that finds a number which, when multiplied by itself, results in a given number. In the context of the Pythagorean Theorem, once you have calculated \(c^2\), taking the square root will allow you to find \(c\), the length of the hypotenuse.
In the example exercise, we arrive at \(c^2 = 625\). To find \(c\), you take the square root of 625. This operation is expressed as: \[c = \sqrt{625}\] This simplifies to \(c = 25\), because \(25 \times 25 = 625\). Understanding how to find square roots is not only important in solving problems involving right triangles but is also essential in higher-level math and real-world applications like calculating distances and dimensions.