   Chapter 7.2, Problem 18E 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

Draw an acute triangle and, by construction, find its orthocentre.

To determine

To draw:

The acute triangle constructs its orthocenter.

Explanation

Procedure used:

STEP 1: We start with the acute triangle ABC.

STEP 2: Set the compasses width to the length of a side of the triangle. Any side will do, but the shortest works with the compasses on B, one end of that line, draw an arc across the opposite side. Label this point F. Repeat for the other end of the line, C. Label this point P.

STEP 3: With the compasses on B, set the compasses width to more than half the distance to P. From B and P, draw two arcs that intersect, creating point Q.

STEP 4: Use a straightedge to draw a line from C to Q. The part of this line inside the triangle forms an altitude of the triangle

STEP 5: Repeat the process with the compasses on C, set the compasses width to more than half the distance to F. From C and F, draw two arcs that intersect, creating point E.

STEP 6: Use a straightedge to draw a line from B to E. The part of this line inside the triangle forms an altitude of the triangle

STEP 7: The point where the two altitudes intersect is the orthocenter of the triangle. The point O is the orthocenter of the acute triangle.

Repeat steps 5, 6 on the third side of the triangle. This will help convince you that all three altitudes do in fact intersect at a single point. But two altitudes are enough to find that point.

Given:

Draw an acute triangle. Construct its orthocenter.

Clarification:

STEP 1: We start with the acute triangle ABC.

STEP 2: Set the compasses width to the length of a side of the triangle. Any side will do, but the shortest works with the compasses on B, one end of that line, draw an arc across the opposite side

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