Circle Clusters Slides CIRCLE
CLUSTERS
Geometric Properties & Algebraic Systems
Advanced Geometry / Algebra II
Warm-up
5 Minutes: Mental or Calculator Challenge
Solve the following system for \(a\), \(b\), and \(c\):
\(a + b = 9\)
\(b + c = 15\)
\(a + c = 14\)
Strategy Hint: Try substitution or adding all three equations together.
Bridging the Gap
Geometry \(\rightarrow\) Algebra
Embedded media
Watch: Problem 3 (15:05 - End)
Critical Observation
How does the external tangency of two circles translate into an addition equation ?
Pause Points
15:05: Predict the variables.16:11: Connect radii to total distances.
The Radii-Sum Theorem
r₁ r₂ d
If two circles are externally tangent, the distance between their centers is:
d = r₁ + r₂
"The spatial boundary becomes an algebraic sum."
CIRCLE CLUSTERS
Main Activity: 25 Minutes
Step 1
Identify the distances between centers in each cluster.
Step 2
Define variables for each radius and set up your system.
Step 3
Solve the system using substitution or matrix methods.
Circle Clusters Worksheet Circle Clusters
Advanced Geometry / Algebra II
Student Name
Date
Warm-up: Systems Review
Solve the following system for \(a\), \(b\), and \(c\). Show your steps.
\[ \begin{cases} a + b = 9 \\ b + c = 15 \\ a + c = 14 \end{cases} \]
Cluster 1: The Triple Threat
A B C 110 125 115
Three circles (\(A, B, C\)) are mutually tangent. The distances between their centers are shown. Define variables for each radius (\(r_A, r_B, r_C\)) and solve for their lengths.
System of Equations
Solution Area
Cluster 2: The Precise Chain
Distances between centers are given to the nearest tenth. Solve the system. Note: Circle \(D\) has a known radius of \(12.4\). Solve for \(r_A, r_B,\) and \(r_C\).
Measurements:
Dist(A to B) = 30.2
Dist(B to C) = 28.5
Dist(C to D) = 22.1
Given: \(r_D = 12.4\)
Show your work
Closure: The Impossible Radius
Design a set of center distances for 3 tangent circles (\(d_{AB}, d_{BC}, d_{AC}\)) that would result in an impossible geometry (where a radius is negative). Identify the distances and explain the geometric reason why they cannot exist.
Proposed Distances
d(A to B) = _____
d(B to C) = _____
d(A to C) = _____
Geometric Explanation
Blueprint Teacher Guide Blueprint Guide
Teacher Facilitation & Answer Key
LESSON: CIRCLE CLUSTERS
Lesson Objectives
Model spatial relationships with linear equations.
Apply systems of equations (3+ variables) to find circle radii.
Evaluate the physical feasibility of geometric solutions (Triangle Inequality).
Discussion Prompts
"Why is the distance between centers exactly \(r_1 + r_2\) for external tangency?"
"If a radius is zero, what does that mean for the 'circle'?"
"How would the equation change if circles were internally tangent?"
Answer Key
Warm-up: Systems Review
System:
Method: Add all: \(2a+2b+2c = 38 \rightarrow a+b+c = 19\).
Solution:
Cluster 1: The Triple Threat
System:
Solution:
Cluster 2: The Precise Chain
Setup:
Final Radii:
Matrix Calculator Guide (TI-84)
Setting up the Matrix:
For Cluster 1: \([1, 1, 0, 110; 1, 0, 1, 125; 0, 1, 1, 115]\)
Press [2nd] [MATRIX] , scroll to EDIT , select [A] .
Set dimensions to 3 x 4 .
Enter coefficients (use 0 for missing variables).
Solving:
Press [2nd] [QUIT] .
Press [2nd] [MATRIX] , scroll to MATH .
Select B: rref( .
Select [A] and press [ENTER] .
Read the right-most column for values of \(r_A, r_B, r_C\).
Closure Example Solution
Example: \(d_{AB}=10, d_{BC}=10, d_{AC}=30\).
