A triangle has side lengths of 6, 8, and 10. Is it a right triangle? Explain (2 points)
The Correct Answer and Explanation is:
To determine whether a triangle with side lengths of 6, 8, and 10 is a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. The Pythagorean theorem can be expressed mathematically as:
[
c^2 = a^2 + b^2
]
Where:
- (c) is the length of the hypotenuse,
- (a) and (b) are the lengths of the other two sides.
In this case, the sides of the triangle are 6, 8, and 10. Here, 10 is the longest side, so we will treat it as (c), and the other sides (a) and (b) will be 6 and 8, respectively.
Now, we can plug these values into the Pythagorean theorem:
[
10^2 = 6^2 + 8^2
]
Calculating each side, we have:
[
10^2 = 100
]
[
6^2 = 36
]
[
8^2 = 64
]
Now, we add (6^2) and (8^2):
[
36 + 64 = 100
]
Since both sides of the equation are equal:
[
100 = 100
]
This confirms that the triangle satisfies the Pythagorean theorem. Therefore, we conclude that the triangle with side lengths of 6, 8, and 10 is indeed a right triangle.
In summary, to determine whether a triangle is a right triangle, we can utilize the Pythagorean theorem. If the square of the longest side is equal to the sum of the squares of the other two sides, the triangle is a right triangle. In this example, since (10^2) is equal to (6^2 + 8^2), we confirm that the triangle is a right triangle, specifically a 6-8-10 triangle, which is a common example of a right triangle.