# 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 CabriJr. Open the file INSCRIB1 by pressing \begin{align*}Y=\end{align*}, selecting Open..., and selecting the file. In PRODUC1, you are given circle \begin{align*}O\end{align*} and two chords \begin{align*}AB\end{align*} and \begin{align*}CD\end{align*} that intersect at point \begin{align*}X\end{align*}. You are also given the lengths \begin{align*}AX\end{align*}, \begin{align*}BX\end{align*}, \begin{align*}CX\end{align*}, and \begin{align*}DX\end{align*}.

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

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

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 \begin{align*}O\end{align*} and two chords \begin{align*}AB\end{align*} and \begin{align*}CD\end{align*} that intersect at point \begin{align*}X\end{align*}. You are also given the lengths \begin{align*}AX\end{align*}, \begin{align*}BX\end{align*}, \begin{align*}CX\end{align*}, and \begin{align*}DX\end{align*}.

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

Position \begin{align*}AX\end{align*} \begin{align*}BX\end{align*} \begin{align*}CX\end{align*} \begin{align*}DX\end{align*} \begin{align*}AX \cdot BX\end{align*} \begin{align*}CX \cdot DX\end{align*}
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5. What do you notice about the products \begin{align*}AX \cdot BX\end{align*} and \begin{align*}CX \cdot DX\end{align*}?

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 \begin{align*}O\end{align*} and two chords \begin{align*}AB\end{align*} and \begin{align*}CD\end{align*} that intersect at point \begin{align*}X\end{align*}. You are also given the lengths \begin{align*}AX, \ CX\end{align*}, and \begin{align*}DX\end{align*}.

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

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