The three points given form a right triangle. Find the...

The three points given form a right triangle. Find the midpoint of the hypotenuse and verify that the midpoint is an equal distance from all three vertices.
$$\begin{aligned}&P_{1}=(6,-6)\\&P_{2}=(-12,18)\\&P_{3}=(20,42)\end{aligned}$$

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Key Concepts

Right Triangle

A triangle with one 90-degree angle, which implies that the two sides forming the right angle are perpendicular. This triangle type follows the Pythagorean theorem, where the square of the hypotenuse equals the sum of the squares of the other two sides.

Midpoint

The midpoint is the point that divides a line segment into two equal parts. It is calculated by taking the average of the corresponding coordinates of the segment’s endpoints, providing symmetry in many geometric constructions.

Hypotenuse

In a right triangle, the hypotenuse is the side opposite the right angle and is the longest side of the triangle. It is crucial because the properties of the triangle, including the use of the Pythagorean theorem and certain circle theorems, revolve around it.

Distance Formula

The distance formula is used to determine the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and is essential in verifying that points are equidistant from a given center.

Circumcenter

The circumcenter is the point equidistant from all vertices of a triangle and is found at the intersection of the perpendicular bisectors of the triangle's sides. In a right triangle, the circumcenter uniquely lies at the midpoint of the hypotenuse, serving as the center of the circumscribed circle.

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