Question

On a coordinate plane, a triangle is shown. It has points (1, 4), (5, negative 2), and (negative 3, negative 2).
Mary thinks the triangle is equilateral. How would you support or dispute her conjecture?

Calculate the slope of all three sides, and check whether the slopes are equal.
Calculate the slope of all three sides, and check whether any slopes are reciprocals.
Use the distance formula to calculate the length of all three sides, and check whether all sides are congruent.
Use the distance formula to calculate the length of all three sides, and check whether any two sides are congruent.

Answer 1

You should use the distance formula and check whether all three sides are congruent.

Compute side lengths:
- AB between (1,4) and (5,−2): sqrt((5−1)^2 + (−2−4)^2) = sqrt(16+36) = sqrt52 = 2√13
- AC between (1,4) and (−3,−2): sqrt((−3−1)^2 + (−2−4)^2) = sqrt(16+36) = sqrt52 = 2√13
- BC between (5,−2) and (−3,−2): sqrt((−3−5)^2 + (−2+2)^2) = sqrt(64+0) = 8

Since the three side lengths are 2√13, 2√13, and 8 (and 8 ≠ 2√13), the triangle is not equilateral. It is isosceles (AB = AC).