What if a line were drawn outside a circle that appeared to touch the circle at only one point? How could you determine if that line were actually a tangent? After completing this Concept, you'll be able to apply theorems to solve tangent problems like this one.
The tangent line and the radius drawn to the point of tangency have a unique relationship. Let’s investigate it here.
Investigation: Tangent Line and Radius Property
Tools needed: compass, ruler, pencil, paper, protractor
- Using your compass, draw a circle. Locate the center and draw a radius. Label the radius , with as the center.
- Draw a tangent line, , where is the point of tangency. To draw a tangent line, take your ruler and line it up with point . Make sure that is the only point on the circle that the line passes through.
- Using your protractor, find .
Tangent to a Circle Theorem: A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.
To prove this theorem, the easiest way to do so is indirectly (proof by contradiction). Also, notice that this theorem uses the words “if and only if,” making it a biconditional statement. Therefore, the converse of this theorem is also true. Now let’s look at two tangent segments, drawn from the same external point. If we were to measure these two segments, we would find that they are equal.
Two Tangents Theorem: If two tangent segments are drawn from the same external point, then the segments are equal.
In is tangent at point . Find . Reduce any radicals.
Solution: Because is tangent, , making a right triangle. We can use the Pythagorean Theorem to find .
Find , in . Round your answer to the nearest hundredth.
Find the perimeter of .
Solution: , and . Therefore, the perimeter of .
We say that is inscribed in . A circle is inscribed in a polygon, if every side of the polygon is tangent to the circle.
Find the value of .
Because and and are tangent to the circle and also congruent. Set and solve for .
Watch this video for help with the Examples above.
A circle is the set of all points that are the same distance away from a specific point, called the center . A radius is the distance from the center to the outer rim of a circle. A diameter is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. A tangent is a line that intersects a circle in exactly one point. The point of tangency is the point where the tangent line touches the circle.
1. Determine if the triangle below is a right triangle. Explain why or why not.
2. Find the distance between the centers of the two circles. Reduce all radicals.
3. If and are the centers and is tangent to both circles, find .
1. To determine if the triangle is a right triangle, use the Pythagorean Theorem. is the longest length, so we will set it equal to in the formula.
is not a right triangle. And, from the converse of the Tangent to a Circle Theorem, is not tangent to .
2. The distance between the two circles is . They are not tangent, however, and . Let’s add , such that is a rectangle. Then, use the Pythagorean Theorem to find .
3. Because is tangent to both circles, it is perpendicular to both radii and and are similar. To find , use the Pythagorean Theorem.
To find , use similar triangles.
Determine whether the given segment is tangent to .
Algebra Connection Find the value of the indicated length(s) in . and are points of tangency. Simplify all radicals.
are points of tangency for
- Is ? Why?
- Find .
- Find .
- Using the trigonometric ratios, find . Round to the nearest tenth of a degree.
- Fill in the blanks in the proof of the Two Tangents Theorem. Given : and with points of tangency at and . and are radii. Prove :
|4.||Definition of perpendicular lines|
|5.||Connecting two existing points|
|6. and are right triangles|
From the above proof, we can also conclude (fill in the blanks):
- is a _____________ (type of quadrilateral).
- The line that connects the ___________ and the external point _________ and .
- Points , and are all points of tangency for the three tangent circles. Explain why .
- Circles tangent at are centered at and . is tangent to both circles at . Find the radius of the smaller circle if , and .
- Four circles are arranged inside an equilateral triangle as shown. If the triangle has sides equal to 16 cm, what is the radius of the bigger circle?
- Circles centered at and are tangent at . Explain why and are collinear.