Question

The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. \(6 \mathrm{ft}, 8 \mathrm{ft}, 9 \mathrm{ft}\)

Short answer

No, the triangle is not a right triangle.

Step-by-step solution

Identify the hypotenuse

To determine if a triangle is a right triangle, identify the longest side as the potential hypotenuse. In this case, among 6 ft, 8 ft, and 9 ft, the longest side is 9 ft. This will be our assumed hypotenuse.

Apply the Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: \(c^2 = a^2 + b^2\). Substitute the sides of the triangle into the equation: \(9^2 = 6^2 + 8^2\).

Calculate the squares of the sides

Compute the squares of each side: \(9^2 = 81\), \(6^2 = 36\), and \(8^2 = 64\).

Verify Pythagorean theorem condition

Add \(6^2\) and \(8^2\) to check if it equals \(9^2\). Calculate: \(36 + 64 = 100\). Since \(81 eq 100\), the given sides do not satisfy the Pythagorean theorem.

Conclusion

Since \(9^2 eq 6^2 + 8^2\), the triangle with sides 6 ft, 8 ft, and 9 ft is not a right triangle.

Key concepts

Pythagorean Theorem

The Pythagorean Theorem is a cornerstone concept in geometry, particularly when dealing with right triangles. Let's break it down: this theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship is expressed using the formula:

• The hypotenuse is denoted as \( c \)
• The other two sides are denoted as \( a \) and \( b \)
• The theorem is written as \( c^2 = a^2 + b^2 \)

This theorem only holds true for right triangles. Therefore, it becomes a very effective tool for verifying if a triangle is right-angled simply by comparing side lengths. If the squares of two shorter sides add up to the square of the longest side, the triangle is right."

Triangle Sides

Triangle sides are fundamental in determining the type of a triangle, whether it's isosceles, scalene, or right-angled. For a right triangle, it's important to arrange the side lengths from smallest to largest to identify which side is potentially the hypotenuse. The peculiar relationships among these sides determine the geometric properties of the triangle. Let's explore these key points:

• Every triangle side plays a distinct role, and the longest side is typically the hypotenuse in a right triangle.
• In the problem given, side lengths are measured in feet: 6 ft, 8 ft, and 9 ft.
• The order of these lengths helps in verifying the type of triangle. Here, 9 ft is the longest and thus is treated as the potential hypotenuse.

Understanding these relationships helps use the Pythagorean Theorem effectively by ensuring we choose correct values for substitution."

Hypotenuse

The hypotenuse is the side of a right triangle opposite the right angle, and it is always the longest side in the triangle. Identifying the hypotenuse correctly is crucial when applying the Pythagorean Theorem because it must be the value represented by \( c \) in the equation \( c^2 = a^2 + b^2 \).

• When given three side lengths, the largest value will indicate the hypotenuse if the triangle is right.
• For example, with sides 6 ft, 8 ft, and 9 ft, 9 ft is the hypotenuse.
• By testing the lengths using the Pythagorean Theorem, we see the essential role this side plays in identifying and confirming the type of triangle.

Always remember, the distinct characteristic of the hypotenuse is that it always stretches across the right angle, cementing its place in triangle geometry and aiding in classification.