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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 $AX \cdot BX$ 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 $x$.

## Date Created:

Feb 23, 2012

Nov 03, 2014
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