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Problem 576

# Construct a line perpendicular to a given line through a given point outside the line.

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Draw the given line AB and the given point C. Using a compass, draw a circle with center C to intersect line AB at points D and E. Next, connect points D and E, and construct a perpendicular bisector FG of segment DE. Extend FG to intersect with point C, forming line CH, which is perpendicular to line AB and passes through point C.
See the step by step solution

## Step 1: Draw the given line and point

Draw the given line segment and label it as line AB. Also, draw the given point and label it as point C.

## Step 2: Draw a circle with radius greater than half the distance of AB

Using a compass, place the needle at point C and draw a circle with a radius greater than half the distance between points A and B. The circle should intersect line AB at two points. Label these points as D and E.

## Step 3: Draw the segment DE

Using a straightedge, connect points D and E to form line segment DE.

## Step 4: Construct the perpendicular bisector of DE

First, draw a circle with center D and radius DE using a compass. Then draw another circle with center E and the same radius. The two circles will intersect at two points. Label these points as F and G. Now, use a straightedge to connect points F and G. The line FG is the perpendicular bisector of the segment DE.

## Step 5: Extend the perpendicular bisector and label the intersection point

Extend the perpendicular bisector FG until it intersects with point C. Label the intersection point as H. Now, you have constructed line CH, which is perpendicular to line AB and passes through the given point C.

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