Geometry
Ch. 10: Circles
10.6: Investigate Segment Lengths
Name: __________________________________________________
Date: ____________________________________________________
Essential Question: What is the relationship between the lengths of segments in a circle?
Step 1: Construct a Circle and chords.
1. Use the circle tool to draw a circle. Use the point tool to plot four points on the circumference of
the circle. Label the points A, B, C, and D.
2. Create chords. Use the arrow to highlight points A and C.
From the Construct Menu choose segment. Repeat this step
but highlight points B and D. Highlight segments AC and BD.
From the Construct Menu choose intersection. Label the
intersection as E.
Step 2: Measure Segments.
1. Use the arrow to highlight points A and E. From the Measure Menu choose distance. Highlight
points E and C. From the Measure Menu choose distance. Repeat this step for segments BE and
ED.
2. Perform Calculations. From the Number Menu choose calculate. Highlight the measurement for
AE and multiply it by the measurement for EC. Next, highlight the measurement for BE and
multiply it by the measurement for ED. Record your products in the table below.
Segment
Product
AE  EC
BE  ED
3. What do you notice about the products you found in step 2.
4. Drag points A, B, C, and D, keeping E inside the circle. What do you notice about the new
products?
5. Make a conjecture and complete the statement below.
If two chords intersect in the interior of a circle, then the products of their lengths are ____________________.
Step 3: Apply it! Use the conjecture you made to solve for x in the circle below.
Step 1: Construct a circle and secants.
1. From the File Menu, choose New Sketch. Use the circle tool to draw a circle. Use the point tool to
create a point outside of the circle, label the point A. Add two more points on the circumference
of the circle. Label the points B and C.
2. Use the arrow tool to highlight points A and B. From the Construct Menu choose segment. Use
the arrow tool to highlight points A and C. From the Construct Menu choose segment.
3. Use the arrow tool to highlight segment AB and the circle. From
the Construct Menu, choose intersections. Repeat this step but
highlight segment AC and the circle. Label the intersections D and E.
Step 2: Measure Segments.
1. Highlight segment AB, from the Measure Menu, choose length. Highlight
points A and D. From the Measure Menu, choose distance.
2. Highlight segment AC, from the Measure Menu choose length. Highlight
points A and E. From the Measure Menu, choose distance.
3. From the Number Menu, choose calculate. Highlight the measure for AB and multiply it by the
measure for AD. Next highlight the measure for AC and multiply it by the measure for AE. Record your
products in the table below.
Segment
Product
AB  AD
AC  AE
4. What do you notice about the products in step 3?
5. Drag point A keeping it outside of the circle. What happens to the products?
6. Make a conjecture and complete the statement below.
If two secants share an ___________________ outside of a circle, then the product of the length of the secant
and the external portion of the secant will be ___________________________.
Step 3: Apply it! Use the conjecture you made to find the value of x in the circle below.
Step 1: Construct a circle, tangent and secant.
1. From the File Menu, choose New Sketch. Use the circle tool to make a circle.
2. Use the point tool to make a point outside of the circle. Label the point A. Use
the point tool to add two additional points to the circle’s circumference. Label
the points B and C.
3. Use the arrow to highlight the center of the circle and point B. From the
Construct
Menu, choose segment. Now you made a radius of the circle. Highlight points A and B.
From the Construct Menu, choose segment. Our goal is to create a Tangent line. Complete the
statement below:
A radius and tangent line of a circle are _____________________________________ lines.
4. Highlight the center of the circle, point B and then point A. From the Measure Menu, choose Angle.
Move Point A until the angle measures as close to 90 degrees as possible.
5. Highlight point A and C. From the construct menu choose segment. You now have made a tangent
and a secant. Highlight segment AC and the circle. From the Construct Menu, choose intersections.
Label the point E.
Step 2: Measure Segments.
7. Highlight segment AB, from the Measure Menu, choose length.
8. Highlight segment AC, from the Measure Menu choose length. Highlight points A and E. From the
Measure Menu, choose distance.
9. From the Number Menu, choose calculate. Highlight the measure for AB and multiply itself. Next
highlight the measure for AC and multiply it by the measure for AE. Record your products in the table
below.
Segment
Product
AB  AB
AC  AE
10. What do you notice about the products in step 9?
11. Make a conjecture and complete the statement below.
If a ____________________ and a __________________ share an endpoint outside a circle, the product of the
length of the secant and the external portion of the secant will equal the tangent’s length
___________________________.
Step 3: Apply it! Use the conjecture you made to find the value of x in the circle below.