To determine whether a triangle with side lengths of 10, 24, and 27 is a right triangle, we can use the Pythagorean Theorem. The Pythagorean Theorem states that for a right triangle with legs of length 'a' and 'b', and hypotenuse 'c', the following equation holds true:
\[c^2 = a^2 + b^2\]
This theorem is only true for right-angled triangles. For any other type of triangle, this equation does not hold.
Let's step through the process:
1. Assign the longest side as the potential hypotenuse 'c', and the other two sides as 'a' and 'b'.
2. Substitute the lengths of the sides into the Pythagorean Theorem.
Here, the longest side is 27, so we'll use this as 'c'. The other sides will be 'a = 10' and 'b = 24'.
\[c^2 = a^2 + b^2\]
Let's calculate:
\[27^2 = 10^2 + 24^2\]
\[729 = 100 + 576\]
\[729 = 676\]
The sum of the squares of sides 'a' and 'b' is not equal to the square of side 'c'. Since the equation does not hold, the triangle with sides of lengths 10, 24, and 27 is not a right triangle.
Therefore, the correct answer is:
B. False
