Question

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.

Explanation:

An equilateral triangle has all three sides of equal length. The slope of a side gives information about its steepness, not its length. So calculating slopes (first two options) won't help in determining if a triangle is equilateral. The last option checks for only two sides being congruent, which would make it isosceles, not necessarily equilateral. The distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) calculates the length of a side between two points \((x_1,y_1)\) and \((x_2,y_2)\). By calculating the length of all three sides using this formula and checking if all are congruent (equal in length), we can confirm if the triangle is equilateral.

Answer:

Use the distance formula to calculate the length of all three sides, and check whether all sides are congruent.