Chapter 10.5, Problem 45E

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6th Edition

Daniel C. Alexander + 1 other

ISBN: 9781285195698

Textbook Problem

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*In Exercises 45 and 46, complete an analytic proof for each theorem.*

The altitudes of a triangle are concurrent.

To determine

**To prove:**

The altitudes of triangle are concurrent.

Explanation

The triangle

Let

If the point of intersection of the altitudes is at the same point, then they are said to be concurrent.

By theorem,

If two lines are perpendicular, then the product of their slopes is

(i.e.) If

Since

(i.e.)

Obviously the altitude

The slope of the line

Using the slope formula and choosing

Thus the slope of the altitude

The equation of the line

Initially substitute

Substitute

The intersection of these altitudes is found by solving the two altitude equation.

On solving,

The coordinates of orthocenter

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