Properties of Tangents
Remember, a tangent line is a line that
touches the circle at only one point.
Segment Lengths of Circles
 Lengths of segments refer to the
distance from one endpoint to
another
 Lengths are never calculated in
degrees
 Lengths are usually notated by a
number/letter in the middle of the
segment
Today in Class:
1. Work with your group to complete your assigned investigation.
2. Raise your hand when you are finished so Ms. Crabb can check
your work.
3. Choose someone to present your findings to the class.
Investigation 1 – Discover
1. In the space below, draw a circle and a tangent line. *Remember a tangent
line only intersects the circle _____________ time(s). Make sure everyone on
you group has a different size circle.
2.
3.
4.
5.
Draw a point where the circle and the tangent line intersect and label it P.
We call point P a point of _________________.
Draw a radius connecting the center of the circle to the point of tangency.
Measure the angle formed by the radius and the tangent line. Write the angle
measure on your diagram.
6. Compare your diagram and angle measure to your group mates. What do
you notice? Does the size of the circle change the measure of the angle?
7. What can we conclude about the radius of a circle and the point of tangency?
Investigation 1 – Generalize
Tangent – Radius Theorem
In a plane, a line is tangent to a circle if and only if the line is _________________________
to a radius of the circle and the point of tangency.
If k is tangent to circle A, then 𝑘 ⊥ ________ .
Investigation 1 – Apply
̅̅̅̅
̅̅̅̅ .
𝐴𝐵 = 8, ̅̅̅̅
𝐴𝐶 = 20. Find 𝐵𝐶
̅̅̅̅
𝑋𝑌 = 5, ̅̅̅̅
𝑌𝑍 = 12, ̅̅̅̅
𝑋𝑍 = 13
̅̅̅̅ is tangent
Determine if 𝑌𝑍
to Circle X.
Investigation 2 – Construct & Discover
1. Open Geometer’s Sketchpad
2. Click the circle button from the toolbar and construct a circle.
3. Click on the letter button from the toolbar and label the center of the circle
C.
4. Click on the point button from the toolbar. Place two points so they will be
located on the same semicircle.
5. Click on the letter button from the toolbar and label one of these points A
and the other B.
6. Using the mouse tool, select points A and C.
(ONLY these two points
should be highlighted.) Go to the menu bar at the top and select Construct >
Segment. Then do the same for points B and C.
7. Select point A and the segment ̅̅̅̅
𝐴𝐶 . Select Construct > Perpendicular Line.
̅̅̅̅ .
Do the same for point B and segment 𝐵𝐶
** Recall that perpendicular means two lines meet at a _____________ angle.
This means you just constructed two ______________ lines.
8. Select the two lines you just constructed. Select Construct > Intersection.
Label this point P.
**We call this point a common external point.
9. Select points A and P. Select Measure > Distance. Do the same for points B
and P.
10. What do you notice about their measurements?
11. Does this work for all circles? (check your group mates’ constructions)
12. Does this work for any pair of tangent lines? (move points A and B around
̅̅̅̅ and 𝐵𝑃
̅̅̅̅)
your circle and see what happens to the measurements of 𝐴𝑃
Investigation 2 – Generalize
Tangent – External Point Theorem
Tangent segments from a common external point
are ____________________.
̅̅̅̅ are tangent to Circle P, then 𝐵𝐷
̅̅̅̅ and 𝐷𝐶
̅̅̅̅ ≅
If 𝐵𝐷
__________.
Investigation 2 – Apply
All segments that appear to be tangent are tangent. Solve for x.