## IGD: Dropping a Perpendicular Line

Grade:

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:

- Construct circle
*P*. Its radius must be large enough so that it intersects line*AB*at two distinct points. - Mark the two points of intersection,
*R*and*S*. - Draw circles
*R*and*S*, each with the same radius. The radius should be large enough so that the two circles intersect. - Label one point of intersection as point
*Q*. - 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*?