Relate central and inscribed angles, chords and tangent lengths with explicit circle hypotheses.
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Try
A central angle of 120° identifies an arc of 120°. An inscribed angle intercepting that same arc, with its vertex on the other part of the circle, is 60°. Identify the intercepted arc before halving: a chord has both a minor and a major arc.
Predict
Predict what happens when the intercepted chord is a diameter. The relevant arc is a semicircle of 180°, so the inscribed angle is 90°. A tangent is perpendicular to the radius at its contact point; this provides right triangles for length arguments outside a circle.
Experiment
In the diagram, move C in your sketch along the upper semicircle while keeping diameter AB fixed. Its angle remains right even when the two chords no longer look symmetric. Draw the center-to-contact radius before trying to calculate a tangent length.
A diameter AB subtends a right angle at C on the circle. The drawing uses an actual circle and equal coordinate scales; C is the top point in this example.
Understand
Worked argument for a diameter: join center O to C. OA=OC=OB are radii, so triangles AOC and BOC are isosceles. Let their angles at C be α and β. Their angles at A and B also equal α and β. The whole triangle ACB therefore has angle sum 2α+2β=180°, giving angle ACB=α+β=90°.
The general inscribed-angle theorem follows by drawing a diameter through the vertex and using the same isosceles reasoning in two pieces, adding or subtracting according to the center’s location. Invalid argument: “Any angle touching the circle is half a central angle.” The vertex must be on the circle and its two rays must intercept the stated arc; the arc used cannot contain that vertex.
Apply
For chord and tangent proofs, label the center, contact points and equal radii. Two equal-looking chords in different circles need not have the same distance from their centers. Keep the same-circle or equal-radius assumption explicit.
Continue
Depth: the perpendicular-bisector theorem for chords yields equal chord–equal distance results by right-triangle congruence. The tangent-radius perpendicularity can be justified by shortest distance from a point to a line: a nonperpendicular radius would make the line pass closer than the radius, cutting the circle twice.
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Practice and reasoning
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An inscribed angle intercepts an arc of measure 146° that does not contain its vertex. Find the angle in degrees.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
From external P, a tangent PT touches a circle centered O. OP=13 and radius OT=5. Find PT.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
An inscribed angle subtending a diameter measures:
Equal chords of the same circle have:
Prove that a line from the center perpendicular to a chord bisects that chord.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
From an external point P, tangents PA and PB touch one circle. Prove PA=PB.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
