10.3: Circle Product Theorems
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
- This activity was written to be explored with the Cabri Jr. application on the TI-84.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
- To download the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=12512 and select PRODUC1-PRODUC5.
Associated Materials
- Student Worksheet: Circle Product Theorems http://www.ck12.org/flexr/chapter/9694, scroll down to the third activity.
- Cabri Jr. Application
- PRODUC1.8xv, PRODUC2.8xv, and PRODUC3.8xv
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 \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 \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 \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
1. Sample answers:
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*} |
---|---|---|---|---|---|---|
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
4. Sample answers:
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*} |
---|---|---|---|---|---|---|
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
7. Sample answers:
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*} |
---|---|---|---|---|---|
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. \begin{align*}\frac{3}{4}\end{align*}
12. \begin{align*}3 \sqrt{13}\end{align*} or 10.817