Translate a circle diagram into one invariant product: secants, tangents, and the lengths that keep multiplying.
ThinkCompareExplain
Today’s route
A question becomes an explanation.
Try
Enter through a problem. A partial attempt counts.
Connect
Use power of a point, secant, tangent 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
From an external point P, a tangent touches a circle at T with PT = 6. A secant through P meets the circle at A and B, with PA = 4. What is PB?
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
01The tangent length is squared, while the secant contributes two distances from P.
02Both expressions measure the same relationship between P and the circle.
03The point P, not the location of a segment on the page, anchors every product.
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.
power of a point
the product tied to a point and a circle
secant
a line that meets a circle twice
tangent
a line that touches a circle once
Use one of these words to explain something you already noticed.
The core idea
Keep the idea. Explain the conditions.
PT2=PA⋅PB
The idea
Translate a circle diagram into one invariant product: secants, tangents, and the lengths that keep multiplying.
What to remember
In a circle diagram, find the point first; then look for the product it controls.
What information does this method keep?
Worked example
One route, one decision at a time
The problem
Chords AB and CD intersect at X. If XA = 3, XB = 8, and XC = 4, find XD.
Predict each next step
01For intersecting chords, the products from X match: XA * XB = XC * XD.
02Substitute the known lengths: 3 * 8 = 4 * XD.
03So 24 = 4 * XD, and XD = 6.
04The diagram's angles justify the product; the arithmetic only finishes it.
Which step needs a reason, not just a calculation?
See the structure
Products survive the diagram
The far distance is the whole secant, measured from P.
Read the representation
For an interior point X, XA * XB = XC * XD. For an exterior point P, PT2 = PA * PB. The common pattern is a circle-centered product attached to one point.
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.
Power of a point
01Use two segments measured from the same point.
02Keep near and far intersections in the product.
03Check whether the point is inside or outside the circle.
Equal segment lengths
01Do not assume two chords are equal because they look parallel.
02A radius is not automatically a secant segment.
03Products, not sums, carry the invariant here.
Which route can you justify? What would you test first?
Reasoning clinic
Find the step that needs repair.
Investigate these concerns
01Do not assume two chords are equal because they look parallel.
02A radius is not automatically a secant segment.
03Products, not sums, carry the invariant here.
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
Hunt for matching products
Your shared task
Sketch several circles with a point inside or outside. Label four segments, then predict which two products should agree before measuring.
Explore
5 min - draw an interior intersection
Compare
5 min - draw two secants from outside
Defend
5 min - explain what stays fixed
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
01Every factor is measured from the same reference point.
02The tangent theorem is the exterior version of the same product idea.
03A scale drawing can suggest the result, but angle relationships justify it.
Which observation explains the mechanism?
Try → explain → check
Your turn to build the route
The problem
Two secants from P meet a circle as P-A-B and P-C-D. If PA = 5, PB = 20, and PC = 4, find PD.
Predict each next step
01Use the exterior secant relation PA * PB = PC * PD.
02Substitute: 5 * 20 = 4 * PD, so 100 = 4 * PD.
03Divide by 4 to get PD = 25.
04The farther point D is not found by adding a fixed amount; it is fixed by the product from P.
Which step needs a reason, not just a calculation?
Convince a skeptic
Turn a pattern into a proof.
The claim
Power is the circle's way of remembering a point.
Build the argument
01Intersecting chords create a pair of similar triangles through vertical and inscribed angles.
02Similarity turns a ratio of sides into XA * XB = XC * XD.
03The tangent-secant diagram gives the same pattern, so the product is independent of the chosen chord or secant.
What does each step rely on? Where are the conditions used?
Stretch together
Take the idea somewhere new.
Transfer challenge
From P, tangent PT = 10 and secant P-A-B has PA = 4 and AB = 21. Find PB.
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
Computational geometry uses circle tests to detect intersections, tangencies, and collision boundaries. The lesson is broader: a well-chosen invariant can replace a complicated picture with one reliable quantity.
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 must both factors in a secant product be measured from the same point?
Question 2
How does the tangent case fit the product pattern?
Question 3
What evidence would tell you that a labeled diagram is impossible?
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
In a circle diagram, find the point first; then look for the product it controls.
Redraw the challenge and solve it two ways: with the product theorem and with a segment-length check.
A good question belongs to everyone in the room.
1 / 21
← → to navigate and reveal · N for notes · F to present · Esc to exit