Final answer:
triangle ABC is a right triangle.
Step-by-step explanation:
To determine if triangle ABC, with vertices A (-5,-7), B(7,-3) and C(4,6), is a right triangle, we need to check if the Pythagorean theorem holds true for the lengths of the sides of the triangle. We first calculate the lengths of the segments AB, BC, and AC using the distance formula (d = √((x2-x1)² + (y2-y1)²)), and then check to see if the square of the longest side equals the sum of the squares of the other two sides.
Let's calculate the length of each side:
- AB = √((7 - (-5))² + ((-3) - (-7))²) = √(144 + 16) = √160
- AC = √((4 - (-5))² + (6 - (-7))²) = √(81 + 169) = √250
- BC = √((4 - 7)² + (6 - (-3))²) = √(9 + 81) = √90
Next, we determine which side is the longest and test the Pythagorean theorem (a² + b² = c²).
Since AC is the longest side, we check if AC² = AB² + BC²:
- (√250)² = (√160)² + (√90)²
- 250 = 160 + 90
- 250 = 250
Since the equation holds true, triangle ABC is indeed a right triangle.