Question

A triangle has vertices (2,−3), (−2, 1), and (1, 2). Are any of the sides congruent?

A) No, they all have different lengths.
B) Yes, the side with vertices (2,−3) and (1, 2) and the side with vertices (−2, 1) and (1, 2) are congruent.
C) Yes, the side with vertices (2,−3) and (−2, 1) and the side with vertices (2,−3) and (1, 2) are congruent.

Answer 1

After using the distance formula to calculate the lengths of the sides of the triangle with given vertices, we find that no two sides are of equal length. Therefore, none of the sides of the triangle are congruent.

Step-by-step explanation:

To determine if any of the sides of the triangle with vertices (2,−3), (−2, 1), and (1, 2) are congruent, we must calculate the lengths of the sides using the distance formula, which is derived from the Pythagorean theorem. The distance formula is given by: d = √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) are the coordinates of the two points.

Let's calculate the length of the sides:

1. Side between (2, −3) and (−2, 1): d = √((−2 - 2)² + (1 - (−3))²) = √((-4)² + 4²) = √(16 + 16) = √32 ≈ 5.66
2. Side between (2, −3) and (1, 2): d = √((1 - 2)² + (2 - (−3))²) = √((-1)² + 5²) = √(1 + 25) = √26 ≈ 5.10
3. Side between (−2, 1) and (1, 2): d = √((1 - (−2))² + (2 - 1)²) = √((3)² + (1)²) = √(9 + 1) = √10 ≈ 3.16

Upon comparing the lengths, we see that no two sides are of equal length, so the answer is A) No, they all have different lengths.