The converse of the Pythagorean theorem is also a true...

The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle. $$(-3,1),(2,-1), \text { and }(6,9)$$

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Key Concepts

Distance Formula

The distance formula is a fundamental tool in coordinate geometry used to calculate the distance between two points in the plane. It is derived from the Pythagorean theorem and gives the length of the segment connecting two points by taking the square root of the sum of the squared differences of their coordinates.

Pythagorean Theorem

The Pythagorean theorem is a key principle in geometry that establishes a relationship among the three sides of a right triangle. It states that in a right triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the remaining two sides, which provides a method to verify distances and relationships in triangles.

Converse of the Pythagorean Theorem

The converse of the Pythagorean theorem is used to determine if a triangle is a right triangle based on the lengths of its sides. It states that if the sum of the squares of two sides of a triangle equals the square of the third side, then the triangle must have a right angle, thereby confirming its classification as a right triangle within the context of Euclidean geometry.

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