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Circles Practice 3: Combined Problems and Proofs

Section A: Combined Reasoning

Question 1mcq
From point PP, PAPA and PBPB are tangents to a circle with centre OO. If ∠APB=60∘\angle APB=60^\circ, what is true about △APB\triangle APB?
A.It is isosceles.
B.It is scalene.
C.It is right-angled at PP.
D.It is not a triangle.
Question 2mcq
In a circle, PTPT is tangent at TT, OT=6OT=6 cm and OP=10OP=10 cm. Which additional fact helps find ∠OPT\angle OPT after finding PTPT?
A.OT⊥PTOT\perp PT
B.OT=PTOT=PT
C.PP lies on the circle
D.OPOP is tangent
Question 3mcq
Which is the best structure for a circle tangent proof?
A.Given, to prove, construction if needed, theorem-based steps, conclusion
B.Only final answer
C.Only redraw the diagram
D.Assume the tangent lengths are equal without proof
Question 4mcq
Which reason is valid in a tangent proof?
A.OA⊥PAOA\perp PA because radius to point of contact is perpendicular to tangent.
B.OA=PAOA=PA because both touch the circle.
C.PA∥PBPA\parallel PB because both are tangents.
D.OPOP is a radius because PP is external.
Question 5mcq
A student says a line is tangent because it is close to the circle in the diagram. What is wrong?
A.A tangent must be proven to touch the circle at exactly one point.
B.Any nearby line is tangent.
C.A tangent must pass through the centre.
D.A tangent must cut the circle twice.

Section B: Proof and Error Analysis

Question 6mcq
A solution uses equal tangents theorem for segments drawn from two different external points. What is the error?
A.Equal tangents theorem applies to two tangents from the same external point.
B.Tangents can never be equal.
C.The theorem applies only to radii.
D.The line must be a secant.
Question 7mcq
In △OPT\triangle OPT, OTOT is radius and PTPT is tangent. If ∠POT=40∘\angle POT=40^\circ, find ∠OPT\angle OPT.
A.40∘40^\circ
B.50∘50^\circ
C.60∘60^\circ
D.90∘90^\circ
Question 8mcq
A tangent length from PP is 1515 cm and OP=17OP=17 cm. What is the radius?
A.44 cm
B.66 cm
C.88 cm
D.1010 cm
Question 9mcq
From PP, tangents PAPA and PBPB are 2x+52x+5 and 4x−94x-9. What is PAPA?
A.1212
B.1717
C.1919
D.2424
Question 10mcq
Assertion: If PAPA and PBPB are tangents from PP, then PA=PBPA=PB. Reason: OA=OBOA=OB and both radii are perpendicular to their tangents, helping prove congruent right triangles. Which is correct?
A.Both are true and the reason explains the assertion.
B.Both are true but the reason does not explain it.
C.Assertion is true but reason is false.
D.Assertion is false but reason is true.