Segment Squad Slides Segment Squad
Cracking the Codes of Circle Blueprinting
The Glossary
Chord
Secant
Tangent
A. Touches the circle at one point.
B. Endpoints are on the circle.
C. Intersects at two points.
Case #1: Chord Crossing
a b c d
The Blueprint Rule:
\(a \cdot b = c \cdot d\)
"The product of the segments of one chord equals the product of the segments of the other."
Case #2: Secant Duo
E Whole W
The Blueprint Rule:
\(W \cdot E = W \cdot E\)
"The product of the whole secant and its external part is constant."
Case #3: Tangent & Secant
T E W
The Blueprint Rule:
\(T^2 = W \cdot E\)
"Tangent squared equals Whole times External."
Mission Review
Question: Which formula applies when the intersection point is INSIDE the circle?
A) \(W \cdot E = W \cdot E\)
B) \(a \cdot b = c \cdot d\)
C) \(T^2 = W \cdot E\)
D) \(a + b = c + d\)
Segment Squad Notes Segment Squad: Blueprint Notes
Mission: Circles & Segments
Agent:
Date:
Briefing: Essential Terms
Chord:
A segment whose endpoints lie on the circle.
Secant:
A line or segment that intersects a circle at exactly points.
Tangent:
A line or segment that touches the circle at exactly point.
External Segment:
The part of a secant that lies the circle.
CASE #1
Intersecting Chords Theorem
a b c d
The Formula:
\(a \cdot \_\_\_\_\_\_\_\_ = \_\_\_\_\_\_\_\_ \cdot d\)
Drafting Check: If a=6, b=2, c=4, solve for d.
CASE #2
Intersecting Secants Theorem
E1 E2 W1
\(W_1 \cdot E_1 = W_2 \cdot E_2\)
Mnemonic: Whole × External = WE!
CASE #3
Tangent-Secant Theorem
T E W
\(T^2 = W \cdot E\)
Tangent Squared = Whole × External
Segment Squad Lab Segment Squad: Lab Work
Mission Part 1: Field Verification & Mastery
UNIT: CIRCLE GEOMETRY
Phase 1: Terminology Match
1. Chord
2. Secant
3. Tangent
4. External Segment
5. Point of Tangency
A. The part of a secant segment that lies outside the circle.
B. A line or segment that intersects a circle at exactly one point.
C. A segment whose endpoints lie on the circle.
D. The exact location where a tangent line meets the circle.
E. A line or segment that intersects a circle at two points.
Phase 2: Quick Mastery Rounds
Round 1: Chord Logic
x 6 4 9
Solve for x. Show your equation setup:
Round 2: Secant Duo
4 6 5 x
Remember: Whole × External = Whole × External
Round 3: Tangent Squared
12 8 x
Apply \(T^2 = W \cdot E\):
Segment Squad Assessment Mission Report: Circle Segments
Unit Assessment: Mastery Level Check
ID: SEG-ASSESS-01
Agent Name
Score
1. If two chords intersect inside a circle, the product of the segments of one chord is ____________ the product of the segments of the other.
A) Half of
B) Double
C) Equal to
D) Greater than
3 8 x 4
2. Find the value of x in the intersecting chord diagram shown.
A) 6
B) 10.6
C) 11
D) 2
Work Space:
3. When two secant segments meet at a point outside a circle, which formula is used?
A) Part × Part = Part × Part
B) Whole × External = Whole × External
C) Tangent² = Whole × External
D) External + Internal = Whole × 2
4. A secant has an external segment of 5 units and an internal segment of 7 units. What is the length of the WHOLE secant?
A) 5
B) 7
C) 12
D) 35
5. If a tangent segment and a secant segment meet outside a circle, the tangent segment squared is equal to:
A) The product of the whole secant and the external part.
B) The product of the internal and external parts.
C) The sum of the whole secant and the external part.
D) Half of the whole secant.
3 4 9 x
6. Solve for x. (Note: External parts are 3 and 4. Internal parts are 9 and x).
A) 5
B) 6.75
C) 9
D) 12
Work Space:
7. Find the length of a tangent segment if the whole secant is 16 and the external segment is 4.
A) 64
B) 20
C) 8
D) 10
8. A chord is divided into segments of 2x and 4. Another chord is divided into segments of 8 and 3. What is the value of x?
A) 3
B) 6
C) 12
D) 2
9. In the formula \(T^2 = W \cdot E\), what does 'W' stand for?
A) Width of the circle
B) Whole secant length
C) Western part of the chord
D) Weight of the segment
10. If two chords intersect and one is divided into segments of 5 and 6, and the other is divided into segments of 3 and y, find y.
A) 8
B) 10
C) 15
D) 2
Segment Squad Answer Key Segment Squad: Teacher Key
Classified Solutions - For Educators Only
Phase 1: Vocabulary
1. Chord: C
2. Secant: E
3. Tangent: B
4. External Segment: A
5. Point of Tangency: D
Phase 2: Quick Mastery
Round 1 (Chords):
\(x \cdot 6 = 4 \cdot 9 \rightarrow \mathbf{x = 6}\)
Round 2 (Secants):
\(4 \cdot (10) = 5 \cdot (5 + x) \rightarrow 40 = 25 + 5x \rightarrow \mathbf{x = 3}\)
Round 3 (Tangents):
\(12^2 = 8 \cdot (8 + x) \rightarrow 144 = 64 + 8x \rightarrow \mathbf{x = 10}\)
Assessment Report Key
1. Chords Product RuleC
6. Secant Diagram SolveA
2. Chord Solve (3x8=4x)A (6)
7. Tangent Solve (x²=16x4)C (8)
3. Secant Formula RuleB
8. Chord Solve (2x*4=8*3)A (3)
4. Whole Secant SumC (12)
9. Formula Symbol 'W'B
5. Tangent Formula RuleA
10. Chord Solve (5x6=3y)B (10)
Strategy Briefing
Case #2/3 Trap: Students often multiply by the internal part instead of the whole. Always ask: "Is your W representing the entire line from start to finish?"
Q6 Hint: Secant 1: \(External = 3, Internal = 9 \rightarrow Whole = 12\). Secant 2: \(External = 4, Internal = x \rightarrow Whole = 4+x\). Equation: \(3(12) = 4(4+x)\). Result: \(36 = 16 + 4x \rightarrow 20 = 4x \rightarrow x=5\).
