Final answer:
After calculating the lengths of segments AB, BC, and AC using the distance formula and applying the Pythagorean theorem, we conclude that the triangle with vertices A(7,10), B(5,3), and C(7,3) is a right triangle, as the Pythagorean relationship holds true.
Step-by-step explanation:
To determine whether the triangle with vertices A(7,10), B(5,3), and C(7,3) is a right triangle, we can use the Pythagorean theorem, which states that for a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b), given by: a² + b² = c².
Firstly, we will calculate the lengths of the segments AB, BC, and AC using the distance formula, which is √((x2-x1)² + (y2-y1)²).
Note that the lengths of BC and AC indicate that these are vertical and horizontal segments, respectively, which means that ∠BAC is a right angle. The triangle will be a right triangle if the following relationship holds using the Pythagorean theorem:
BC² + AC² = AB²
2² + 7² ≡ √53²
4 + 49 = 53
53 = 53
Since the relationship holds true, we can conclude that the triangle with the given vertices is indeed a right triangle.