Circle Secrets Slides Circle Secrets
Segment Relationships & Constructions
Tools of the Trade
The Compass
Creates the circle and ensures a constant radius from point O.
The Straightedge
Draws the secant lines that cut through the circle.
"Geometric construction is the bridge between pure logic and visual reality."
Anatomy of a Secant
P A B C D
The Whole Segment
Length PB or PD
The External Part
Length PA or PC
The Outside-In Theorem
If two secant segments share an endpoint outside a circle, then:
\[ \text{whole}_1 \cdot \text{outside}_1 = \text{whole}_2 \cdot \text{outside}_2 \]
Construction Guide
1
Circle O
Mark O and sweep your compass 360°.
2
Point P
Pick any point outside the circle boundary.
3
Two Secants
Draw two lines from P through the circle.
Let's Calculate
P 4 6 5 x
The Equation:
\[ 4 \cdot (4 + 6) = 5 \cdot (5 + x) \]
"Can you solve for x? Hint: Simplify the parentheses first!"
Perfect Circles Guide Perfect Circles Guide
Construction & Segment Theory
Name: ________________________
Date: _________________________
The Secant-Secant Power Theorem
If two secant segments are drawn from an external point P , the product of the external part and the whole length is equal for both paths.
PA · PB = PC · PD
A B C D P
Step-by-Step Construction
1
Establish the Circle
Mark a center point O . Set your compass and sweep a full circle.
O
2
Exterior Point P
Choose any point P outside the circle. This will be the vertex for both secants.
P
3
Draw and Label Secants
Use a straightedge to draw two rays from P through the circle. Label the four intersection points.
P A B C D
Drafting Zone
Practice your construction here. Follow the steps above using your compass and straightedge.
Segment Showdown Worksheet Segment Showdown
Applying the Secant-Secant Theorem
Name: ________________________
Date: _________________________
Part 1: Drafting
Construct Circle O and an exterior point P . Draw two secant segments. Measure your segments in cm and verify the theorem: (Outside) × (Whole) = (Outside) × (Whole).
Product 1: ________
Product 2: ________
Part 2: Calculation
Solve for x in each figure. Show your algebraic work clearly in the boxes.
6 10 8 x
Algebraic Work:
5 7 4 x
Algebraic Work:
x 12 9 11
Algebraic Work:
Challenge Problem
Secant 1 from P has an external part of 3 and an internal chord of 15 .
Secant 2 from P has a total length of 18 . Solve for its external part (x ).
Algebraic Work:
Circle Master Key Circle Master Key
Teacher Reference & Solutions
Master Guide
Part 1: Drafting Reference
Ensure students use compasses for circles. Their products should be equal (within ±5% measurement error). Example: If external is 4cm and whole is 10cm, the other secant product must also be ≈ 40.
Part 2: Algebraic Solutions
1
Equation: 6 · (6 + 10) = 8 · (8 + x)
Whole 1: 16 | Whole 2: 8 + x
6 10
6(16) = 8(8 + x)
96 = 64 + 8x → 32 = 8x
x = 4
2
Equation: 5 · (5 + 7) = 4 · (4 + x)
Whole 1: 12 | Whole 2: 4 + x
5 7
5(12) = 4(4 + x)
60 = 16 + 4x → 44 = 4x
x = 11
3
Equation: x · (x + 12) = 9 · (9 + 11)
Whole 1: x + 12 | Whole 2: 20
x
x² + 12x = 9(20) → x² + 12x - 180 = 0
Quad. Formula: [-12 ± √(144 + 720)] / 2
x ≈ 8.70 (Positive root only)
4
Equation: 3 · (3 + 15) = x · 18
Whole 1: 18 | Whole 2: 18
54 = 18x
x = 3
The external part must be 3 to maintain the product of 54.
Pedagogical Note: In Problem 3, students often struggle with the quadratic equation. If they haven't learned it yet, you can encourage them to use 'Guess and Check' or a graphing calculator to find the positive value of x .
