10.3: Circle Product Theorems
This activity is intended to supplement Geometry, Chapter 9, Lesson 6.
Problem 1 – Chord-Chord Product Theorem
Start the Cabri Jr. application by pressing the APPS key and selecting CabriJr. Open the file INSCRIB1 by pressing , selecting Open..., and selecting the file. In PRODUC1, you are given circle and two chords and that intersect at point . You are also given the lengths , , , and .
1. Move point to four different points and collect the data in the table below and calculate the products and .
Position | ||||||
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1 | ||||||
2 | ||||||
3 | ||||||
4 |
2. What do you notice about the products and ?
3. If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is ____________ to the product of the lengths of the segments of the other chord.
Problem 2 – Secant-Secant Product Theorem
Open the file PRODUC2. You are given circle and two chords and that intersect at point . You are also given the lengths , , , and .
4. Move point to four different points and collect the data in the table below and calculate the products and .
Position | ||||||
---|---|---|---|---|---|---|
1 | ||||||
2 | ||||||
3 | ||||||
4 |
5. What do you notice about the products and ?
6. If two secant segments share the same endpoint outside of a circle, then the product of the lengths of one secant segment and its external segment ___________ the product of the lengths of the other secant segment and its external segment.
Problem 3 – Secant-Tangent Product Theorem
Open the file PRODUC3. You are given circle and two chords and that intersect at point . You are also given the lengths , and .
7. Move point to four different points and collect the data in the table below and calculate and .
Position | |||||
---|---|---|---|---|---|
1 | |||||
2 | |||||
3 | |||||
4 |
8. What do you notice about the products and ?
9. If a secant segment and a tangent segment share an endpoint outside of a circle, then the product of the lengths of the secant segment and its external segment _________ the square of the length of the tangent segment.
Problem 4 – Application of Product Theorems
10. Find the value of .
11. Find the value of .
12. Find the value of .