Question

practice using coordinate algebra to solve problems.
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:

Step1: Recall the property of an equilateral triangle

An equilateral triangle has all three sides congruent.

Step2: Analyze each option

• Option 1: Slope measures steepness. Equal slopes mean parallel lines, not equal - length sides.
• Option 2: Reciprocal slopes (if positive - reciprocal, perpendicular; if negative - reciprocal, not a property of equilateral triangle) have no relation to side - length equality.
• Option 3: The distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) calculates side lengths. Checking if all three sides are congruent (using distance formula) is the correct way to verify an equilateral triangle.
• Option 4: Checking only two sides congruent would verify an isosceles triangle, not necessarily equilateral.

Answer:

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