Interactive Geometry Dictionary
How Do You Construct Perpendicular Lines?

Constructing a Perpendicular Line at a Point on a Line (interactively!)

Construct a line perpendicular to line AB at point P on the line using the applet below.

• Step 1. Construct a circle with center P.
• Step 2. Mark the two points R and S of intersection of this circle with line AB.
• Step 3. Construct two congruent circles, centered at points R and S, with radii large enough so that the two circles intersect.
• Step 4. Mark one of the points of intersection Q.
• Step 5. Connect the point of intersection Q to point P; this line is perpendicular to line AB and passes through P.

Use the Measure button to see if you have constructed a perpendicular line.

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You want to prove that angles QPR and QPS in the applet below are both right angles.

• Triangles QPR and QPS are congruent. Why?
• What can you say about angles QPR and QPS?

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Answer

Triangles QPR and QPS are congruent by Side-Side-Side; sides PR and PS are radii of the circle centered at P, sides QR and QS are radii of congruent circles, and side QP is shared. Since angles QPR and QPS are corresponding angles of congruent triangles, they have the same measure; they are also supplementary. Thus, angles QPR and QPS must both be right angles.

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