Question

(Lesson 9-4) Is a triangle with measures 18,24 , and 30 a right triangle? (Lesson 6-6)

Short answer

Yes, a triangle with measures 18, 24, and 30 is a right triangle.

Step-by-step solution

Identify the Sides

Identify the sides of the triangle based on the measures given: 18, 24, and 30.

Determine the Hypotenuse

The largest side will act as the hypotenuse in a right triangle. Here, 30 is the largest measure and will be considered the hypotenuse.

Apply the Pythagorean Theorem

According to the Pythagorean theorem, in a right triangle with legs of lengths \(a\) and \(b\), and hypotenuse \(c\), it holds that \(a^2 + b^2 = c^2\). For this triangle, use \(a = 18\), \(b = 24\), and \(c = 30\). Substitute into the formula: \(18^2 + 24^2 = 30^2\).

Perform Calculations

Calculate each side of the equation. Compute \(18^2 = 324\), \(24^2 = 576\), and \(30^2 = 900\). Add the squares of the two legs: \(324 + 576 = 900\).

Verify Equality

Verify if the sum of the squares of the legs equals the square of the hypotenuse: \(324 + 576 = 900\). Since both sides of the equation are equal (900 = 900), the condition of the Pythagorean theorem is satisfied.

Key concepts

Pythagorean Theorem

The Pythagorean Theorem is an essential mathematical principle, especially when dealing with right triangles. This theorem outlines the relationship between the lengths of the sides in a right triangle. It states that for any right triangle, the sum of the squares of the two shorter sides (known as the legs) is equal to the square of the longest side, which we call the hypotenuse. In formula form, this is represented as:

• \(a^2 + b^2 = c^2\)

Here, \(a\) and \(b\) denote the legs, and \(c\) is the hypotenuse. Applying this theorem helps us to test whether a given set of measures can form a right triangle.
When you encounter a triangle problem where you need to determine if it's a right triangle, always look to the Pythagorean Theorem as your key tool. Understanding how the sum of squares equates will give insights into the nature of the triangle at hand.

Triangle Sides

When examining triangle sides to determine if they form a right triangle using the Pythagorean Theorem, you must first identify which side is the hypotenuse. This requires knowing all three sides of the triangle.
The triangle sides can be labeled as follows:

• The two shorter sides are referred to as the legs.
• The longest side is known as the hypotenuse.

In our example, the side lengths provided are 18, 24, and 30. Among these, 30 is the longest, and thus, it is the hypotenuse. This classification is crucial, as it aids us in correctly applying the theorem.
Proper identification of the hypotenuse and legs ensures the correct calculations when testing the validity of the Pythagorean Theorem. By squaring the length of the legs and ensuring their sum matches the hypotenuse squared, we confirm if the sides can construct a right triangle.

Hypotenuse Determination

Understanding how to accurately determine the hypotenuse is vital because it impacts how you apply the Pythagorean Theorem. Remember, the hypotenuse is always the side opposite the right angle and the longest side in a right triangle.
To identify the hypotenuse among a set of sides:

• Compare the lengths of all sides given.
• Select the largest measure as the hypotenuse.

In the exercise where the side measures are 18, 24, and 30, 30 is the hypotenuse because it is the largest.
This step is crucial because, when applying the Pythagorean Theorem, the hypotenuse is the side that must be squared separately and balanced with the combined squares of the other two sides. Failing to correctly determine the hypotenuse may lead to incorrect conclusions about the nature of the triangle.