# Constructing Tangents to Circles

## Objective

SWBAT construct tangents to circles at a specified point and from a point outside the circle and explain why the constructions work.

#### Big Idea

More than just recipes...Constructions are more like proofs, and in this lesson, students will need to understand the reasons why they work.

## Activating Prior Knowledge

25 minutes

In order to construct tangents to circles students will need to know some things and they will know how to do some things as well. For example, they'll need to know that a tangent to a circle is perpendicular to the radius drawn to the point of tangency. They will also need to know how to construct the perpendicular to a line at a point. Finally, they'll need to know and believe that the midpoint of the hypotenuse of a right triangle is equidistant from the triangle's vertices.

I start by telling students to write down as fact that any tangent drawn to a circle is perpendicular to the radius drawn to the point of tangency.

Next, I have students practice a construction with which they are very familiar: the perpendicular bisector of a line segment. After that, I show them how to construct the perpendicular to a segment at a point that is not the midpoint of the segment. Finally, I show them how to construct the perpendicular to a line segment through one of the segment's endpoints. For these constructions, I give each student a copy of Constructing Perpendiculars.

Next, I explain to my students that the construction we'll be doing in this lesson relies on an important fact about right triangles: The midpoint of the hypotenuse is equidistant from the vertices of the triangle. I offer two demonstrations to convince students that this is a true statement. See the following screencasts to get a feel for how these demonstrations go.

## Construction 1: Tangent to a Circle at a Point

15 minutes

In this section students will be constructing the tangent to a circle at a specified point of tangency. I'll give each student a copy of Construction 1_Tangent through point on circle. This construction is almost directly related to the constructions we did in the Activating Prior Knowledge section so I expect that most students will be able to work through the construction on their own. I do emphasize, however, to students that they are responsible for detailing the steps of the construction and giving the geometric basis for why these steps achieve the desired results.

When students have finished their constructions and write-ups, I'll have a few students present their work. Again, since the construction is fairly straightforward, my focus will be on precisely describing the steps of the construction and explaining why these steps achieve the desired result.

## Construction 2: Tangent from a Point Outside the Circle

25 minutes

In this section, students will be constructing the tangent from a specified point outside the circle. I will be walking students through the overall plan and rationale for this construction. I perform the construction on sketchpad, which will give students a big picture view of the construction, and I explain why the steps I'm taking achieve the desired result. See the following screencast to get a glimpse of how that demonstration and explanation goes.

Once I've given the demonstration, it will be time for students to complete Construction_Tangent from an Outside Point. I don't hand this out until after the demonstration because I want students to be focused on the demonstration and not be trying to simultaneously complete the construction. I also want some delay between the demonstration and their completing the construction to make sure that they are actually thinking as they perform the construction and not just mimicking steps.

As in the previous section, I remind students that they will need to detail the steps of the construction and explain why they achieve the desired results.

When students have finished, I will collect the papers and take them home to read and provide feedback. Students will need to complete this proof and explain why it works on the upcoming unit test so it is important for them each to know how to perform and explain the construction after this lesson.