Problem 36 - Prealgebra

Question

If \(c\) is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. (lesson \(9-4\) ) $$a=?, b=27, c=82$$

Step-by-Step Solution

Answer

The missing measure \(a\) is approximately 77.4.

Step 1: Recall the Pythagorean Theorem

The Pythagorean Theorem relates the lengths of the sides of a right triangle. Given a triangle with sides of length \(a\), \(b\), and hypotenuse \(c\), the theorem is given by: \(a^2 + b^2 = c^2\). In this case, we need to find \(a\).

Step 2: Substitute the Known Values

Plug in the known values into the Pythagorean Theorem: \(a^2 + 27^2 = 82^2\).

Step 3: Calculate 82 Squared

Compute \(82^2\): \(82 \times 82 = 6724\).

Step 4: Calculate 27 Squared

Compute \(27^2\): \(27 \times 27 = 729\).

Step 5: Solve for \(a^2\)

Substitute the values from steps 3 and 4 into the equation and solve for \(a^2\): \(a^2 + 729 = 6724\).Subtract 729 from both sides: \(a^2 = 6724 - 729 = 5995\).

Step 6: Solve for \(a\)

Find the square root of 5995: \(a = \sqrt{5995}\).

Step 7: Approximate the Result

Calculate \(a\) to the nearest tenth: \(a \approx 77.4\), using a calculator to determine the precise square root.

Key Concepts

Right Triangle

A right triangle is a fundamental concept in geometry and forms the basis for the Pythagorean Theorem. It has one angle that is exactly 90 degrees, known as the right angle. The sides forming this right angle are called the 'legs' of the triangle, while the opposite side is the longest side, termed the 'hypotenuse'.

• A right triangle helps us explore relationships between side lengths.
• The Pythagorean Theorem is used specifically for right triangles.

Understanding the properties of right triangles is crucial when using the Pythagorean Theorem, as it applies only to these types.

Hypotenuse

The hypotenuse is the diagonal side opposite the right angle in a right triangle. It is always the longest side of the triangle because it spans across the largest area.

• In any right triangle, the hypotenuse is always greater than either of the legs.
• It is essential in calculations involving the Pythagorean Theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

When calculating values, knowing which side is the hypotenuse helps provide clarity and accuracy in solving problems.

Solving for a Side

In problems involving right triangles, we often need to find the length of an unknown side. The Pythagorean Theorem, given as \(a^2 + b^2 = c^2\), is pivotal for this process, where \(c\) is the measure of the hypotenuse, and \(a\) and \(b\) are the legs.

• First, substitute the known values into the equation.
• In this case, known values are \(b=27\) and \(c=82\).
• The equation becomes \(a^2 + 27^2 = 82^2\).

This setup allows us to isolate and solve for the unknown side, enhancing our understanding and application of mathematical relationships.

Square Root Calculation

After solving for \(a^2\), we need to calculate \(a\) by taking the square root. This is crucial because it transforms the squared result into the actual length value we are seeking.

• Calculate the square of the hypotenuse and the known leg first.
• Solve for \(a^2\): subtract \(27^2 = 729\) from \(82^2 = 6724\).
• We find \(a^2 = 5995\).
• Find \(a\) by computing \(\sqrt{5995}\), which approximates to 77.4.

Using square root calculations is essential in finalizing measurements, providing the precise length of the unknown side.