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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.

ID: 12512

Time Required: 20 minutes

## Activity Overview

Students will use dynamic models to find patterns. These patterns are the Chord-Chord, Secant-Secant, and Secant-Tangent Theorems.

Topic: Circles

• Chord-Chord, Secant-Secant, and the Secant-Tangent Product Theorems

Teacher Preparation and Notes

Associated Materials

## Problem 1 – Chord-Chord Product Theorem

Students will begin this activity by investigating the intersection of two chords and the product of the length of the segments of one chord and the product of the length of the segments of the other chord.

Students will be asked to collect data by moving point A\begin{align*}A\end{align*}. Students are asked to calculate the products by hand on their accompanying worksheet. Students are asked several questions about the relationship among the products.

As an extension, prove the chord-chord product theorem using similar triangles.

## Problem 2 – Secant-Secant Product Theorem

Students will investigate the intersection of two secants and the product of the lengths of one secant segment and its external segment and the product of the lengths of the other secant segment and its external segment.

Students will be asked to collect data by moving point A\begin{align*}A\end{align*}. They are to calculate the products by hand on their accompanying worksheet. Students are asked several questions about the relationship among the products.

As an extension, prove the secant-secant product theorem using similar triangles.

## Problem 3 – Secant-Tangent Product Theorem

Students will investigate the intersection of the product of the lengths of one secant segment and its external segment and the square of the tangent segment.

Students will be asked to collect data by moving point A\begin{align*}A\end{align*}. Students are asked to calculate the products by hand on their accompanying worksheet. Students are asked several questions about the relationship among the products.

As an extension, prove the secant-tangent product theorem using similar triangles.

## Problem 4 – Application of the Product Theorems

Students will be asked to apply what they learned in Problems 1–3 to solve a few problems.

## Solutions

Position AX\begin{align*}AX\end{align*} BX\begin{align*}BX\end{align*} CX\begin{align*}CX\end{align*} DX\begin{align*}DX\end{align*} AXBX\begin{align*}AX \cdot BX\end{align*} CXDX\begin{align*}CX \cdot DX\end{align*}
1 2.70 0.99 1.51 1.78 2.67 2.68
2 2.51 1.05 1.38 1.90 2.63 2.62
3 2.93 0.91 1.80 1.49 2.67 2.68
4 1.80 1.08 2.51 0.77 1.94 1.93

2. They are equal.

3. equal

Position AX\begin{align*}AX\end{align*} BX\begin{align*}BX\end{align*} CX\begin{align*}CX\end{align*} DX\begin{align*}DX\end{align*} AXBX\begin{align*}AX \cdot BX\end{align*} CXDX\begin{align*}CX \cdot DX\end{align*}
1 5.94 3.82 6.65 3.42 22.69 22.74
2 4.85 3.16 5.85 2.62 15.33 15.33
3 4.03 2.75 5.32 2.09 11.08 11.12
4 7.47 4.96 7.92 4.68 37.05 37.07

5. They are equal.

6. equals

Position AX\begin{align*}AX\end{align*} CX\begin{align*}CX\end{align*} DX\begin{align*}DX\end{align*} AX2\begin{align*}AX^2\end{align*} CXDX\begin{align*}CX \cdot DX\end{align*}
1 4.22 6.13 2.90 17.81 17.77
2 3.42 5.40 2.16 11.70 11.66
3 2.45 4.56 1.32 6.00 6.02
4 9.46 11.21 7.98 89.49 89.46

8. They are equal.

9. equals

10. 6

11. 34\begin{align*}\frac{3}{4}\end{align*}

12. 313\begin{align*}3 \sqrt{13}\end{align*} or 10.817

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