   Chapter 10.5, Problem 43E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

For ∆ A B C , the vertices are A 0 ,   0 , B a ,   0 , and C b ,   c . In terms of a , b , and c , find the coordinates of the orthocenter of ∆ A B C . (The orthocenter is the point of concurrence for the altitudes of a triangle.)

To determine

To find:

The coordinates of the orthocenter of ABC in terms of a, b, and c with the vertices A0, 0, Ba, 0, and Cb, c.

Explanation

The triangle ABC with vertices A0, 0, Ba, 0, and Cb, c.is shown in the above figure.

Let AJ-, BK-, CH- be the altitudes of the triangle ABC.

The point of intersection of the altitudes is the orthocentre.

By theorem,

If two lines are perpendicular, then the product of their slopes is -1 or one slope is negative reciprocal of the other slope.

(i.e.) If l1l2, then m1.m2=-1 or m2=-1m1

Since AB- is the horizontal line, slope is zero.

(i.e.) mAB-=0

Obviously the altitude CH- is vertical and it has the equation,

x=b

The slope of the line BC- with vertices Ba, 0, and Cb, c is as follows.

Using the slope formula and choosing x1=a, x2=b, y1=0, and y2=c

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