Let right angles, supplementary angles, and intersecting chords turn a circle diagram into equations.
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Today’s route
A question becomes an explanation.
Try
Enter through a problem. A partial attempt counts.
Connect
Use chord, cyclic, power to name what you notice.
Convince
Build an argument a partner can check.
The goal is a method someone else can use.
The opener
Start with a question.
Opening problem
AB is a diameter of a circle with AB = 10. Point C lies on the circle and AC = 6. Find BC.
Think · 2 min
Sketch, test a small case, or make a first guess.
Pair · 3 min
Compare methods. Find one thing you agree on and one question.
Be ready to share your route, even if it is unfinished.
Compare before checking
Put two attempts in conversation.
Explain
Tell a partner how you started. Let them finish before responding.
Question
Which step feels certain? Which step still needs evidence?
Build
Combine one useful idea from each attempt into a new route.
Can you explain your partner’s method in your own words?
Board conversation
Notice before naming.
Look for structure
01A diameter creates a right angle at every point on the circle.
02Opposite angles in a cyclic quadrilateral add to 180 degrees.
03A product of two lengths often replaces a difficult angle chase.
Make a conjecture
We think ___ because ___. A case that might break this is ___.
What changed? What stayed the same?
Useful language
Now the language has a job.
chord
a segment whose endpoints lie on the circle
cyclic
four or more points that lie on one circle
power
a product such as PA·PB that stays fixed for a point P
Use one of these words to explain something you already noticed.
The core idea
Keep the idea. Explain the conditions.
EA⋅EB=EC⋅ED
The idea
Let right angles, supplementary angles, and intersecting chords turn a circle diagram into equations.
What to remember
Circle geometry becomes manageable when you translate the visible condition into the right invariant.
What information does this method keep?
Worked example
One route, one decision at a time
The problem
From an external point P, one secant meets a circle at A and B with PA = 3 and PB = 12. A second secant meets it at C and D with PC = 4. Find PD.
Predict each next step
01Both lines start at the same external point P.
02The secant-secant theorem gives PA x PB = PC x PD.
03Substitute: 3 x 12 = 4 x PD, so 36 = 4PD.
04Therefore PD = 9; the farther intersection is D, so 9 is consistent with PC < PD.
Which step needs a reason, not just a calculation?
See the structure
One point, two secants, one conserved product
Measure both distances along each secant from the same point P.
Read the representation
Although the two secants cut the circle at different locations, the product of near and far distances from P is the same.
Look together
Point to the part that explains the relationship. Change one input and predict what changes.
What can you see here that was harder to see in words?
Two approaches
Compare the reasoning.
Angle route
01Use inscribed angles and cyclic quadrilaterals.
02Best when angles or a right triangle are visible.
03Watch orientation and supplementary pairs.
Product route
01Use tangent-secant or chord products.
02Best when several lengths are given.
03Check which point is near and which is far.
Which route can you justify? What would you test first?
Reasoning clinic
Find the step that needs repair.
Investigate these concerns
01Use tangent-secant or chord products.
02Best when several lengths are given.
03Check which point is near and which is far.
Repair together
Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.
A useful correction explains why the original step fails.
Work together
Sort the circle clues
Your shared task
Groups classify diagram clues as diameter, cyclic-angle, tangent, or chord-product information before solving.
Explore
5 min - label each clue
Compare
6 min - choose a theorem
Defend
4 min - compare two solution paths
Everyone should be able to explain the group’s final claim.
Share → connect
What did the room discover?
Bring your evidence
Share a conjecture, a useful example, or a route that changed.
Connections to investigate
01A theorem is a translation rule: the diagram supplies the conditions, and the rule supplies the equation.
02Right angles from diameters often finish a problem with one Pythagorean calculation.
03Power-of-a-point products work because the circle condition is shared, not because the two lines look symmetric.
Which observation explains the mechanism?
Try → explain → check
Your turn to build the route
The problem
From an external point P, a tangent has length PT and a secant has near and far distances PA = 5 and PB = 20. Find PT.
Predict each next step
01The tangent-secant theorem says PT2 = PA x PB.
02Substitute the known distances: PT2 = 5 x 20 = 100.
03Lengths are positive, so take the positive square root.
04Thus PT = 10.
Which step needs a reason, not just a calculation?
Convince a skeptic
Turn a pattern into a proof.
The claim
If two chords intersect inside a circle at E, then EA · EB = EC · ED.
Build the argument
01Join A to C and B to D; the intersecting chords create vertical angles at E.
02Equal inscribed angles give two similar triangles, so EA/ED = EC/EB.
03Cross-multiply the proportion to obtain EA x EB = EC x ED.
What does each step rely on? Where are the conditions used?
Stretch together
Take the idea somewhere new.
Transfer challenge
Two chords AB and CD intersect at E. If AE = 3, EB = 8, and CE = 4, find ED.
Choose a route
Name the objects, the conditions, and a relationship that might help. Start with one test case.
What transfers from the earlier example? What is different?
Peer review
Check the argument together.
Restate
Explain the other person’s route before evaluating it.
Test
Try a small case or boundary case. Does every step still make sense?
Revise
Fix an assumption, add a reason, or find a second method.
Where did your thinking change?
Beyond this problem
Where might this idea help?
A connection
The same circle relationships power optical designs, radar ranging, geometric constructions, and coordinate proofs where a product is easier to preserve than an angle.
Discuss the model
01What objects or quantities are represented?
02Which relationship from today is preserved?
03What assumption would you check before trusting it?
Invent one more situation with the same structure.
Exit ticket
Leave with an explanation.
Question 1
Why does a diameter create a right angle on the circle?
Question 2
When should a secant problem become a product equation?
Question 3
What must be true before you apply the intersecting-chords theorem?
Choose one. Write an answer with a reason, then compare with a partner.
Reflect together
Make your thinking visible.
I used to think…
Name a claim or approach that changed.
Now I can explain…
Give the reason, not only the result.
I still wonder…
Write a question worth bringing to the next circle.
Share one sentence. Ask one follow-up question.
The idea to keep
Circle geometry becomes manageable when you translate the visible condition into the right invariant.
On the practice set, mark the exact segment or angle that triggers each theorem before calculating.
A good question belongs to everyone in the room.
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