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Problem 618

# Using a straight edge only, construct the perpendicular to a given diameter of a circle from any point off the diameter.

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To construct the perpendicular to a given diameter of a circle from any point off the diameter using a straight edge only, follow these steps: 1. Draw the circle and its diameter, labeling endpoints as A and B. 2. Choose a point outside the diameter, label it as C. 3. Connect C to A and C to B using straight edge. 4. Construct midpoints M and N of segments CA and CB, respectively. 5. Draw a line through points M and N, extending beyond the circle. 6. Find the intersection point D of this line with the circle. 7. Connect point C to D, which is the perpendicular to diameter AB.
See the step by step solution

## Step 1: Draw the circle and diameter

First, draw a circle and the diameter of the circle. Label the diameter as AB, where A and B are the endpoints of the diameter.

## Step 2: Select a point outside the diameter

Choose a point outside the diameter of the circle; label this point as C.

## Step 3: Connect C to A and B

Using the straight edge, draw a line segment connecting point C to point A and another line segment connecting point C to point B.

## Step 4: Construct midpoints

Construct the midpoint of line segment CA and the midpoint of line segment CB. Label the midpoints as M and N, respectively.

## Step 5: Draw lines through the midpoints

Using the straight edge, draw a line through points M and N, extending beyond the circle.

## Step 6: Find the intersection point

Identify the intersection point of the line through points M and N with the original circle. Label this point as D.

## Step 7: Draw the perpendicular to the diameter

Using the straight edge, draw the line segment connecting point C to point D. This line segment CD is the required perpendicular to the diameter AB. In conclusion, by following these steps and using a straight edge only, we were able to construct the perpendicular to a given diameter of a circle from a point outside the diameter.

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