Math Problem Statement

Select the correct answer.

The points D(-2,2), E(2,5), and F(5,1) form △DEF in the coordinate plane. What condition verifies that the triangle is a right triangle?

 A. 

The product of the slopes of two of the sides is -1.

 B. 

The product of the slopes of two of the sides is 1.

 C. 

Two of the sides of the triangle are equal.

 D. 

Three of the sides of the triangle are equal.

Solution

The correct answer is:

A. The product of the slopes of two of the sides is -1.

Explanation:

To determine if a triangle is a right triangle, one way is to check if two of its sides are perpendicular. In the coordinate plane, two lines are perpendicular if the product of their slopes is -1. This confirms that the triangle has a right angle, which is the defining condition for a right triangle.

Would you like further details on how to calculate slopes and verify this?

Here are 5 related questions to expand on this information:

  1. How can you find the slope between any two points on a coordinate plane?
  2. How do you verify if two lines are perpendicular on a coordinate plane?
  3. Can a triangle with equal sides be a right triangle? Why or why not?
  4. What is the distance formula for finding the lengths of sides in the coordinate plane?
  5. How can the Pythagorean Theorem be used to confirm if a triangle is right?

Tip: When verifying a right triangle on a coordinate plane, always check for perpendicular slopes or use the Pythagorean Theorem if side lengths are known.

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Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Slope of a Line
Right Triangles

Formulas

Slope formula: (y2 - y1) / (x2 - x1)
Condition for perpendicular slopes: m1 * m2 = -1

Theorems

Perpendicular Slope Theorem

Suitable Grade Level

Grades 8-10

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