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10.3: Circle Product Theorems

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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 Cabri Jr. Open the file INSCRIB1 by pressing Y=, selecting Open..., and selecting the file. In PRODUC1, you are given circle O and two chords AB and CD that intersect at point X. You are also given the lengths AX, BX, CX, and DX.

1. Move point A to four different points and collect the data in the table below and calculate the products AX \cdot BX and CX \cdot DX.

Position AX BX CX DX AX \cdot BX CX \cdot DX
1
2
3
4

2. What do you notice about the products AX \cdot BX and CX \cdot DX?

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 O and two chords AB and CD that intersect at point X. You are also given the lengths AX, BX, CX, and DX.

4. Move point A to four different points and collect the data in the table below and calculate the products AX \cdot BX and CX \cdot DX.

Position AX BX CX DX AX \cdot BX CX \cdot DX
1
2
3
4

5. What do you notice about the products AX \cdot BX and CX \cdot DX?

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 O and two chords AB and CD that intersect at point X. You are also given the lengths AX, \ CX, and DX.

7. Move point A to four different points and collect the data in the table below and calculate AX^2 and CX \cdot DX.

Position AX CX DX AX^2 CX \cdot DX
1
2
3
4

8. What do you notice about the products AX^2 and CX \cdot DX?

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 x.

11. Find the value of x.

12. Find the value of x.

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Feb 23, 2012

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Aug 19, 2014
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