How to Construct a Tangent of a Circle

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  • 0:02 Intro to Geometric…
  • 0:53 Interior Tangent of a Circle
  • 1:33 Exterior Tangent of a Circle
  • 2:42 Tangent Through Two Circles
  • 4:17 Lesson Summary
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Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

It's pretty amazing what you can draw with just a straight edge and a compass! In this lesson, you'll learn how you can draw tangents of a circle, whether you are given a point on the circle or outside of the circle and if you have two circles.

Intro to Geometric Construction

Back in the day, before people had these sophisticated and cool computer programs that let them draw all kinds of stuff (even 3D stuff), people used just three tools: a compass, a straight edge, and a pencil. If you don't count the pencil, it's only two tools! This form of drawing that doesn't use numbers is called geometric construction. In this lesson, you'll learn how you can draw tangents of a circle such as these:

construct circle tangent

A tangent to a circle is the line that touches the edge of the circle at a single point. In the above picture, you can see three different kinds of tangents. You have the tangent that passes through a particular point on the circle, the tangent that passes through a particular point outside the circle, and you have the tangent that touches two circles.

Let's go ahead and learn how to draw each of these tangents.

Tangent Passing Through a Point On a Circle

1. To draw a tangent through a given point on the circle, you first draw a line from the center of your circle passing through the point on the circle.

2. Then, set your compass to a width smaller than your radius, and draw two arcs on the line you just drew on either side of your point on the circle.

3. Now, set your compass to the width from your one point to one of the arcs you just drew. Place one end of your compass on one of the arcs and draw an arc on one side of the line. Then, place your compass on the other arc, and draw an arc that intersects with the other arc you just drew.

4. Your tangent line can now be drawn by drawing a line through your intersection and the point on the circle.

Tangent Passing Through a Point Outside of a Circle

If you are given a point outside the circle, you'll draw your tangent in a slightly different way that involves constructing parallel lines.

1. First, connect your outside point to the center of your circle.

2. Then, you'll need to find the midpoint of this line. To do this, set your compass to a width that is more than half the length of your line but no longer than your line. Then, set your compass on one end of your line, and draw an arc on either side of the line. Move your compass to the other end of your line, and draw arcs that intersect with the arcs you just drew.

3. To find your midpoint, place your straight edge so it connects your two intersections. Where your straight edge intersects your line is where your midpoint is. Mark your midpoint.

4. Next, take your compass and change the width so it is the same distance from your external point to the midpoint. Then place your compass at the midpoint and draw two arcs that intersect your circle above and below your line.

5. Your tangent line can then be drawn by connecting your external point to either of the arcs that intersect your circle. Whenever you draw a tangent of a circle given an external point, you'll always end up with two tangents.

Tangent Touching Two Circles

When you have two circles that your tangent needs to touch, the construction process is actually a bit easier.

1. You'll first need to construct two parallel lines that pass through the center of each circle. To do this, draw a line that passes through one of the circles.

2. Then, draw a line that connects the centers of your circles.

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