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10.3: Circle Product Theorems

Difficulty Level: At Grade Created by: CK-12

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 AXBX and CXDX.

Position AX BX CX DX AXBX CXDX
1
2
3
4

2. What do you notice about the products AXBX and CXDX?

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 AXBX and CXDX.

Position AX BX CX DX AXBX CXDX
1
2
3
4

5. What do you notice about the products AXBX and CXDX?

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 AX2 and CXDX.

Position AX CX \begin{align*}DX\end{align*} \begin{align*}AX^2\end{align*} \begin{align*}CX \cdot DX\end{align*}
1
2
3
4

8. What do you notice about the products \begin{align*}AX^2\end{align*} and \begin{align*}CX \cdot DX\end{align*}?

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 \begin{align*}x\end{align*}.

11. Find the value of \begin{align*}x\end{align*}.

12. Find the value of \begin{align*}x\end{align*}.

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TI.MAT.ENG.SE.1.Geometry.10.3

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