Drop a perpendicular line to line AB from a point P not on the line using the applet below.

• Step 1. Construct a circle with center P. Its radius must be large enough so that it intersects line AB at two distinct points.
• Step 2. Mark the two points of intersection.
• Step 3. Draw two circles with the same radius, one centered at each point of intersection marked in Step 2. The radius should be large enough so that the two circles intersect each other.
• Step 4. Mark one of the two points of intersection of the two circles just constructed.
• Step 5. Draw a line through the point of intersection marked in Step 4 and point P. This line passes through point P and is perpendicular to line AB.

Use the Measure button to make sure you have constructed a perpendicular line.

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A Closer Look at Dropping a Perpendicular —Why Does it Work?

To prove that the construction given above works, you must show that the line through point P and the intersection of the circles is perpendicular to the given line. Answer the following questions using the applet below to prove this. You can drag many of the points to see if the relationships you find will always hold.

• Triangles PQR and PQS are congruent. Why?
• What can you conclude about angles RQT and SQT? Explain.
• What can you say about triangles QTR and QTS? Explain.

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Answer

Triangles PQR and PQS are congruent by Side-Side-Side; two corresponding sides are radii of the circle centered at P; two more corresponding sides are radii of congruent circles; and the third side PQ is shared. Angles RQT and SQT are congruent because they are supplements of corresponding angle of the congruent triangles PQR and PQS. Triangles QTR and QTS are congruent by Side-Angle-Side. Sides QR and QS are radii of congruent circles, side QT is shared, and you just proved that angles RQT and SQT are congruent. You can conclude that angles QTR and QTS are both right angles because they are corresponding angles of congruent triangles and they are supplementary to one another.

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