## IGD: Dropping a Perpendicular Line

9-12
Standards:
Math Content:
Geometry

Constructing a line perpendicular to a given line and through a point not on the line is one of the most important constructions in geometry. This process is known as dropping a perpendicular, because a perpendicular line is "dropped" from the point to the line.

Drag points A, B, and P to change the arrangement of the points.

Use the buttons, in order, to show how to drop a perpendicular line. (See the Exploration section for details.) The Measure button can be used to measure the angle formed by the original line and the line constructed.

The Start Over button will reset the entire drawing.

To drop a perpendicular line, perform the following steps:

1. Construct circle P. Its radius must be large enough so that it intersects line AB at two distinct points.
2. Mark the two points of intersection, R and S.
3. Draw circles R and S, each with the same radius. The radius should be large enough so that the two circles intersect.
4. Label one point of intersection as point Q.
5. Draw line PQ. This line passes through point P and is perpendicular to AB.

To prove that this construction works, show that PQ is perpendicular to the given line. Click the Why It Works button to draw in some segments that will help with the proof.

• Triangles PQR and PQS are congruent. Can you explain why?
• What can you conclude about ∠RQT and ∠SQT? Explain.
• Triangles QTR and QTS are also congruent. Can you explain why?
• Because triangles QTR and QTS are congruent and because R and S lie on the same line, what can you conclude about ∠QTR and ∠QTS?