Use coordinates to prove that the midpoint of the hypotenuse...

Key Concepts

Coordinate Geometry

Coordinate geometry involves representing geometric figures using a coordinate system, allowing algebraic methods to solve geometric problems. In this proof, it is used to assign coordinates to the vertices of the triangle so that algebraic formulas can be applied to determine distances and midpoints.

Distance Formula

The distance formula, derived from the Pythagorean theorem, computes the distance between two points in a coordinate system by taking the square root of the sum of the squares of the differences of their coordinates. This formula is crucial in proving that the midpoint of the hypotenuse is equidistant from all vertices by calculating distances between points.

Midpoint Formula

The midpoint formula is used to find the point that is exactly halfway between two given points. In the context of the problem, it is applied to determine the midpoint of the hypotenuse, which is then shown to be equidistant from the triangle's vertices.

Right Triangle Properties

Right triangle properties especially relate to the fact that in a right triangle, the hypotenuse is the longest side and the right angle creates specific relationships between the sides. One key property is that the circumcenter, or the center of the circle passing through all three vertices, lies at the midpoint of the hypotenuse, which is demonstrated using coordinate methods.

Transcript

00:01 Our goal in this problem is to prove that the midpoint of the hypotenuse of a right triangle is equidistant from all three of the vertices.

00:08 Now, by definition, we already know that it's equidistant from two, the endpoints of hypotenuse.

00:13 So really, we just need to prove that the third is equivalent to that, but we'll show all three.

00:17 Now, of simplicity's sake, we'll position one of our vertices, specifically the one with the right angle, at the origin.

00:29 And then we'll position one side along the x -axis and the other along the y -axis.

00:49 The midpoint that we're looking for will be somewhere right here.

00:52 We'll call that maybe point p.

00:55 And that will have coordinates just by the midpoint formula of a plus 0 over 2.

01:03 The midpoint of ab, a plus 0 over 2, comma, b plus 0 over 2, or that's just a over 2, b over 2.

01:16 So we need to do this, we need to show that this is the same distance away from a, b, and c.

01:21 So the distance from a to p, we have a minus, minus a over two squared plus b over two minus zero squared.

01:41 Now a notice is 2a over 2.

01:43 So this is 2a minus a over 2 squared plus just b over 2 squared.

01:52 So that's the square root of a over 2 squared plus b over 2 squared, which works out to the square root of a squared plus b squared over 4.

02:12 Or bringing the four out, since we can take the square root of it, one half times the square root of a squared plus b squared.

02:24 So that's the distance from a to p...

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