Question

If \(c\) is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. \(a=30, c=34, b=?\)

Short answer

The missing measure \(b\) is 16.

Step-by-step solution

Identify the Pythagorean Theorem

The Pythagorean Theorem is given by the equation \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse of a right triangle, and \(a\) and \(b\) are the other two sides.

Substitute Known Values

Substitute the given values \(a = 30\) and \(c = 34\) into the Pythagorean Theorem: \(30^2 + b^2 = 34^2\).

Calculate Squares

Calculate the squares of 30 and 34: \(30^2 = 900\) and \(34^2 = 1156\). Substitute these results into the equation to get \(900 + b^2 = 1156\).

Solve for \(b^2\)

Subtract 900 from both sides of the equation to isolate \(b^2\): \(b^2 = 1156 - 900\). This simplifies to \(b^2 = 256\).

Find \(b\)

Take the square root of both sides to solve for \(b\): \(b = \sqrt{256}\). Therefore, \(b = 16\).

Key concepts

Understanding the Hypotenuse

In any right triangle, the side opposite the right angle is known as the hypotenuse. This side is always the longest of the triangle's three sides. The hypotenuse plays a key role in trigonometry and geometry because it is the basis for the Pythagorean Theorem.
For example, if you're given a triangle where the hypotenuse is labeled as 'c', it is crucial to remember that it is always the side that you solve for when using the Pythagorean Theorem. This theorem, written as \(a^2 + b^2 = c^2\), identifies the mathematical relationship between all three sides of a right triangle.
Key things to remember about the hypotenuse:

• It is opposite the right angle.
• It is the longest side in a right triangle.
• It is represented as 'c' in the Pythagorean Theorem.

Exploring the Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This specific angle makes the right triangle very useful in various areas of mathematics, especially in geometry and trigonometry.
In a right triangle, the two sides that form the right angle are known as the legs, and they are typically labeled as 'a' and 'b'. The right angle lends itself to several important properties and formulas, such as the Pythagorean Theorem, which is fundamental for working with right triangles.
The relevance of right triangles in real-world applications includes:

• Designing and building structures to ensure stability and strength.
• Calculating distances and angles in navigation and surveying.
• Solving trigonometry problems to find unknown lengths or angles.

Demystifying the Square Root

The square root is the inverse operation of squaring a number. In simple terms, if squaring a number results in a value, taking the square root reverses that process to find the original number.
In the context of the Pythagorean Theorem, once we isolate \(b^2\), the square root is used to find the length of side 'b'. For example, \(b^2 = 256\) means that the square root of 256 will give the value of \(b\). Hence, \(b = \sqrt{256} = 16\).
Understanding square roots is important because:

• They help in solving quadratic equations.
• They are used in determining distances between two points in coordinate geometry.
• They are fundamental for understanding and simplifying radical expressions.