Question

The coordinates of the vertices of ABC are A(1,2), B(-5,3), and C(-6,-3). Prove that ABC is .

a) ABC is not isosceles.
b) ABC is isosceles.
c) ABC is equilateral.
d) ABC is a right triangle.

Answer 1

After calculating the distances between the vertices of triangle ABC, we find that AB = BC, which means that triangle ABC is isosceles. It is not equilateral because AC has a different length, and it is not a right triangle as the Pythagorean theorem does not hold for the lengths of the sides.

Step-by-step explanation:

To determine the type of triangle ABC with vertices A(1,2), B(-5,3), and C(-6,-3), we must calculate the distances between each pair of vertices (the lengths of the sides) using the distance formula: √((x2 - x1)² + (y2 - y1)²). After calculating, we find:

• AB = √((-5 - 1)² + (3 - 2)²) = √(36 + 1) = √37
• AC = √((-6 - 1)² + (-3 - 2)²) = √(49 + 25) = √74
• BC = √((-6 + 5)² + (-3 - 3)²) = √(1 + 36) = √37

Since AB = BC, triangle ABC is isosceles. It is not equilateral because AC has a different length. To check for a right triangle, we use the Pythagorean theorem on the two equal sides and the third side, but in this case, √37 + √37 is not equal to √74, so triangle ABC is not a right triangle.