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.
$$\begin{aligned}&P_{1}=(-2,1)\\&P_{2}=(6,-5)\\&P_{3}=(2,-7)\end{aligned}$$

Key Concepts

Circumcenter in Right Triangles

The circumcenter of a triangle is the point that is equidistant from all of its vertices. In a right triangle, this point is uniquely located at the midpoint of the hypotenuse. This property is significant in coordinate geometry as it allows for a straightforward method to find a point that balances the triangle, which can have applications in various geometric and analytical contexts.

Midpoint Formula

The midpoint formula is used to determine the center point of a line segment defined by two endpoints in the coordinate plane. It involves averaging the x-coordinates and the y-coordinates of the two endpoints, resulting in the coordinates of the midpoint. This concept is essential when locating a central or balancing point on a segment.

Distance Formula

Derived from the Pythagorean theorem, the distance formula calculates the straight-line distance between two points in a coordinate plane. By computing the square root of the sum of the squares of the differences in the x-coordinates and y-coordinates, this formula verifies the equality of distances which is crucial in problems involving equidistant points, such as verifying the circumcenter of a triangle.

Right Triangle Properties

In right triangles, certain unique properties apply due to the presence of a 90-degree angle. One notable property is that the circumcenter (the point equidistant from all three vertices) lies at the midpoint of the hypotenuse, simplifying the process of finding a point that is equally distant from all vertices.

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