Hello, Aqua!

I have a proof. .I hope it is what you want.

In triangle ABC, a line connects C directly with the base AB

to form a 90° angle.

They now want us to prove this (why it forms a 90° angle)

by using Analytical geometry. Code:

| C
| *(p,q)
| *: *
| * : *
| * : *
| * : *
| * : *
| * : *
|* : *
A * - - - * - - - - - - - * B
(0,0) (x,0) (r,0)
| D

Place side AC on the x-axis with vertex A at the origin.

Let vertex B have coordinates (r,0), and vertex C (p,q).

Draw a line from vertex V to any point D on side AC: .D(x,0).

. . and we will minimize the length of CD.

To make it easier, we consider the quantity, y = CD².

. . Note that, if CD² is a minimum, then CD is a minimum.

Using the Distance Formula, we have: .y .= .(x - p)² + q²

. . which simplifies to: .y .= .x² - 2px + p² + q²

This is an up-opening parabola; its minimum is at its __vertex__.

. . The vertex of a parabola is at: .x = -b/2a

We have: .a = 1, b = -2p. .Hence: .x .= .(-2p)/-2 .= .p

So distance CD is a minimum when point D is (p,0).

This means that point D is __directly__ __below__ point C.

. . Hence: .__/__CDA = __/__CDB = 90°

Therefore, CD is an *altitude* to side AB.