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.
$$\begin{aligned}&P_{1}=(0,-5)\\&P_{2}=(-6,4)\\&P_{3}=(6,-1)\end{aligned}$$

Key Concepts

Midpoint Formula

The midpoint formula is used to find the point that is exactly halfway between two given points in the Cartesian plane. This is calculated by averaging the x-coordinates and the y-coordinates of the two points, resulting in the point ((x1 + x2)/2, (y1 + y2)/2). This concept is crucial for determining the center of a line segment, such as the hypotenuse in a right triangle.

Distance Formula

The distance formula calculates the distance between two points in the coordinate plane using the square root of the sum of the squares of the differences in their x and y coordinates. Expressed as ?((x2 - x1)² + (y2 - y1)²), it is essential for verifying that a particular point is equidistant from other points, such as confirming that a midpoint is also the circumcenter of a triangle.

Circumcenter of a Triangle

The circumcenter is the point in a triangle that is equidistant from all three vertices and serves as the center of the circle that can pass through these vertices. In the special case of a right triangle, the circumcenter is located at the midpoint of the hypotenuse, a property that simplifies the process of finding the circle's center and the equidistant verification.

Properties of Right Triangles

Right triangles have specific attributes that distinguish them from other triangles, one of which is the location of the circumcenter. Since the hypotenuse acts as the diameter of the circumscribed circle in a right triangle, its midpoint is inherently equidistant from all three vertices. This property is particularly useful in problems that involve finding midpoints and verifying equidistance among points.

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