Problem 42 - Precalculus

Question

Find the area of each triangle with measures given. $$a=9, b=40, c=41$$

Step-by-Step Solution

Answer

The area of the triangle is 180 square units.

Step 1: Verify Right Triangle

First, check if the given triangle is a right triangle using the Pythagorean theorem. For a triangle with sides \(a\), \(b\), and \(c\), if \(a^2 + b^2 = c^2\), then the triangle is right-angled. Here, calculate \(9^2 + 40^2 = 81 + 1600 = 1681\). Also, calculate \(41^2 = 1681\). Since these are equal, the triangle is a right triangle.

Step 2: Use Right Triangle Area Formula

Since we have identified that the triangle is a right triangle, use the formula for the area of a right triangle: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). In this case, consider \(a = 9\) as the base and \(b = 40\) as the height. Substitute these values into the formula: \[\text{Area} = \frac{1}{2} \times 9 \times 40\].

Step 3: Calculate the Area

Now, perform the calculation: \(\frac{1}{2} \times 9 \times 40 = \frac{1}{2} \times 360 = 180\). Therefore, the area of the triangle is 180 square units.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental rule in geometry for working with right triangles. It helps you determine whether a triangle is a right triangle if you know its side lengths. In a triangle, if the square of one side (the hypotenuse) is equal to the sum of the squares of the other two sides, then the triangle is right-angled. This is expressed mathematically as:
\[a^2 + b^2 = c^2\]
where \(a\) and \(b\) are the legs and \(c\) is the hypotenuse. For our triangle with sides \(a = 9\), \(b = 40\), and \(c = 41\), we check:

• \(9^2 + 40^2 = 81 + 1600 = 1681\)
• \(41^2 = 1681\)

Since these values are the same, the theorem confirms this triangle is a right triangle. Understanding this theorem is crucial because it allows us to identify the type of triangle based on its side lengths.

Triangle Area Formula

The formula to find the area of any triangle involves its base and height. However, for a right triangle, this formula simplifies due to its unique properties.
The area of a right triangle can be found using:
\[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]
In this particular formula:

• The base and height are the two legs \(a\) and \(b\) of the right triangle.
• The hypotenuse \(c\) is not used in the area calculation.

In our exercise, the values \(a = 9\) and \(b = 40\) represent the base and the height. Plug these into the formula to find the area:

• \(\text{Area} = \frac{1}{2} \times 9 \times 40\)
• The calculation gives \(180\) square units.

This straightforward application of the formula provides a reliable way to find the area of any right triangle.

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. The side opposite the right angle is the longest and is called the hypotenuse. In right triangles, the two other sides are known as "legs."
Right triangles have special properties that make solving them simpler than other triangles. With a right triangle, many calculations, like finding the height or the area, utilize the Pythagorean Theorem or specific formulas related to their structure.
In our scenario, with side lengths \(a = 9\), \(b = 40\), and \(c = 41\):

• \(c\) is the hypotenuse, as it's the longest side.
• \(a\) and \(b\) function as the legs.

This configuration allows the use of straightforward formulas for calculations, ensuring efficient problem-solving. Understanding these properties of right triangles is essential for anyone learning fundamental geometry.