# High School: Geometry

### Circles HSG-C.A.4

4. Construct a tangent line from a point outside a given circle to the circle.

It's construction time, so tell your students to put on their hard hats. Actually, don't. You'll only get groans and eye rolls.

Students should already know that constructions involve straightedges and compasses rather than jackhammers and drills. Well, it depends on what you mean by drills. Hopefully they've become adept enough in using these instruments because they'll have to put them both to use.

If students don't already know the properties of circles and tangents before this construction, they should take away a few main points from it:

• Tangents drawn to a circle are perpendicular to the circle's radius at the point of tangency.
• Two tangents drawn to a circle from the same point outside the circle are equal. You can have students do this construction and measure the segments from the point of tangency to the shared point.
• Two tangents drawn to a circle from the same point outside the circle make an angle that, when bisected, includes the circles center. Students can construct the angle bisector and see for themselves.
• Tangents to a circle at either end of a diameter are parallel.

Given a circle with center B and a point A outside the circle, the construction of a tangent to ⊙B that goes through A is relatively simple. What's important is that students also understand the properties of circles and their tangents in order to make these constructions.