Question

If \(c\) is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. \(a=6, b=3, c=?\)

Short answer

The hypotenuse \(c\) is approximately 6.7.

Step-by-step solution

Recall the Pythagorean Theorem

The Pythagorean Theorem is a fundamental relation in geometry among the three sides of a right triangle. It states that for a right triangle with legs of lengths \(a\) and \(b\), and hypotenuse length \(c\), the formula is: \(a^2 + b^2 = c^2\).

Substitute known values into the formula

We are given that \(a = 6\) and \(b = 3\). Substitute these values into the Pythagorean Theorem: \(6^2 + 3^2 = c^2\).

Calculate the squares of the legs

First, calculate \(6^2\) which equals 36, and \(3^2\) which equals 9. Substitute these into the equation to get: \(36 + 9 = c^2\).

Add the squares of the legs

Add the numbers you obtained: \(36 + 9 = 45\). This means that \(c^2 = 45\).

Solve for the hypotenuse length \(c\)

To find \(c\), take the square root of 45: \(c = \sqrt{45}\).

Simplify the square root

\(\sqrt{45}\) can be simplified to \(\sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\).

Approximate the value

Calculate \(3\sqrt{5}\) by approximating \(\sqrt{5} \approx 2.236\), so \(3 \times 2.236 = 6.708\). Round to the nearest tenth to get \(c \approx 6.7\).

Key concepts

Understanding Right Triangles

Right triangles are a fundamental concept in geometry. They are called "right" because one of their angles is exactly 90 degrees, forming an "L" shape. This right angle creates a natural connection among the triangle's sides, known as the legs and the hypotenuse. The two sides that form the right angle are referred to as the "legs." They can vary in length but must meet at that 90-degree corner.
The longest side of a right triangle, opposite the right angle, is known as the "hypotenuse." It is always the side you calculate using the Pythagorean Theorem when its length is missing.

• The hypotenuse is distinctively the diagonal line stretching between the two legs.
• In our particular example, the legs are lengths 6 and 3, while the hypotenuse is represented by the letter "c."

Right triangles are often used in various applications, including designing, construction, and solving complex mathematical problems. Understanding the basic properties of a right triangle is vital for applying the Pythagorean Theorem effectively.

Hypotenuse Calculation with the Pythagorean Theorem

The Pythagorean Theorem is an essential tool in calculating the hypotenuse of a right triangle. This theorem provides a clear relationship: the square of the hypotenuse equals the sum of the squares of the two other sides. In mathematical terms, it is presented as: \(a^2 + b^2 = c^2\).
In our problem, you have been provided with the lengths of the legs: \(a = 6\) and \(b = 3\). Here's how you can calculate the hypotenuse step-by-step:

• The first step is to square each leg: \(6^2 = 36\) and \(3^2 = 9\).
• Then, add these squares together: \(36 + 9 = 45\).
• Finally, solve for the hypotenuse \(c\) by taking the square root of 45.

This series of steps leads to \(c = \sqrt{45}\), and simplifying this gives \(3\sqrt{5}\). Approximating further, the hypotenuse length is roughly 6.7 units when rounded to the nearest tenth.

Approaching Geometry Problem Solving

Solving geometry problems, especially those involving right triangles, can appear daunting at first. However, with a systematic approach, it becomes manageable and even straightforward. Here are some tips when tackling similar problems:

• Always start by identifying the type of triangle and the given components. Right triangles will directly lead you to consider using the Pythagorean Theorem.
• Write down all known values clearly. This organizational step prevents confusion later on.
• Follow the procedure of theorems step-by-step, as seen in our example, where each operation — squaring, adding, taking roots — makes logical sense in sequence.
• When a root is not a clear number, simplify it as much as possible and then approximate without losing accuracy. Calculators are handy for this final step.

Remember that in geometry, visualization helps a lot. Sketching the triangle also clarifies the problem and often guides the solution process. With practice and methodical strategies, solving these problems becomes intuitive.