• Apr 10th 2007, 12:39 AM
Aqua
I really hope someone can help with this.

I am afrikaans so please excuse my english names.

We have a project in Maths asking us the following:

In a rectangle ABC, where a line connects C directly with the basis with AB to form a 90 degrees angle(i believe its called a hightline, in direct translation).

http://img74.imageshack.us/img74/4767/gfnm1.png

They now want us to prove this(Why it forms a 90 degree angle) by using Analytical geomatry, can someone help out 'n bit please?
• Apr 10th 2007, 06:57 AM
Soroban
Hello, Aqua!

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

Quote:

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.