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Circles Final Assessment

Section A: Core Concepts

Question 1mcq
A segment joining the centre of a circle to a point on the circle is called:
A.radius
B.chord
C.secant
D.tangent
Question 2mcq
A tangent to a circle has how many points of contact with the circle?
A.One
B.Two
C.Three
D.Infinitely many
Question 3mcq
If OTOT is radius and PTPT is tangent at TT, then:
A.OT⊥PTOT\perp PT
B.OT∥PTOT\parallel PT
C.OT=PTOT=PT
D.O,T,PO,T,P are collinear
Question 4mcq
Which fact is used to prove tangent-radius perpendicular theorem?
A.The perpendicular from the centre to a line gives the shortest distance.
B.All chords are equal.
C.A tangent passes through centre.
D.Every radius is a tangent.
Question 5mcq
In right triangle OTPOTP, OT⊥PTOT\perp PT. If ∠OPT=25∘\angle OPT=25^\circ, find ∠POT\angle POT.
A.25∘25^\circ
B.55∘55^\circ
C.65∘65^\circ
D.90∘90^\circ
Question 6mcq
From external point PP, OP=15OP=15 cm and radius OT=9OT=9 cm. Find tangent length PTPT.
A.99 cm
B.1010 cm
C.1212 cm
D.1515 cm

Section B: Theorem Applications

Question 7mcq
Tangents drawn from the same external point to a circle are:
A.equal in length
B.parallel always
C.perpendicular to each other always
D.diameters
Question 8mcq
Which congruence idea is commonly used to prove equal tangents from an external point?
A.RHS congruence of right triangles
B.BPT
C.AA similarity only
D.Midpoint theorem
Question 9mcq
From point PP, PAPA and PBPB are tangents. If PA=18PA=18 cm, find PBPB.
A.99 cm
B.1212 cm
C.1818 cm
D.3636 cm
Question 10mcq
From vertex AA of a tangential quadrilateral, the two tangent segments are 66 cm each. From vertex BB, the two tangent segments are 44 cm each. What is the total of these four segments?
A.1010 cm
B.1616 cm
C.2020 cm
D.2424 cm
Question 11mcq
If PAPA and PBPB are tangents from PP, why is △APB\triangle APB isosceles?
A.PA=PBPA=PB
B.OA=OBOA=OB only
C.OPOP is a tangent
D.ABAB is a diameter
Question 12mcq
Which statement is acceptable in a tangent proof?
A.OT⊥PTOT\perp PT because tangent at TT is perpendicular to radius OTOT.
B.OT=PTOT=PT because both meet at TT.
C.PP is centre because tangents start at PP.
D.PTPT is radius because it touches the circle.

Section C: Proofs and Perimeters

Question 13mcq
A student assumes PA=QBPA=QB because both are tangents, but they come from different external points PP and QQ. What is wrong?
A.Equal tangents must be from the same external point.
B.Tangents are never equal.
C.Tangents must be radii.
D.The circle must be a triangle.
Question 14mcq
If OT⊥PTOT\perp PT and ∠POT=30∘\angle POT=30^\circ, what is ∠OPT\angle OPT?
A.30∘30^\circ
B.45∘45^\circ
C.60∘60^\circ
D.90∘90^\circ
Question 15mcq
The radius is 1212 cm and tangent length from PP is 1616 cm. Find OPOP.
A.1818 cm
B.2020 cm
C.2424 cm
D.2828 cm
Question 16mcq
Tangents from PP are 3x−13x-1 and 2x+62x+6. Find the tangent length.
A.1717
B.1818
C.1919
D.2020
Question 17mcq
A triangle has an incircle. From a vertex, the two tangent segments to the incircle are 55 cm and xx cm. Find xx.
A.33
B.55
C.1010
D.Cannot be found
Question 18mcq
From PP, PAPA and PBPB are tangents and ∠APB=70∘\angle APB=70^\circ. If OPOP bisects the angle between equal tangents, find ∠APO\angle APO.
A.25∘25^\circ
B.35∘35^\circ
C.45∘45^\circ
D.70∘70^\circ

Section D: Mixed Challenge

Question 19mcq
Which pair correctly names circle elements?
A.Chord: joins two points on circle; tangent: touches at one point
B.Radius: joins two circle points; secant: touches at one point
C.Diameter: outside line; tangent: centre line
D.Chord: passes through centre always; radius: outside segment
Question 20mcq
A line touches the circle at AA, but the solution uses point BB as point of contact. What issue will this cause?
A.The radius used for perpendicularity may be wrong.
B.All points on tangent are contact points.
C.The tangent becomes a diameter.
D.No theorem can ever be used.
Question 21mcq
In proving PA=PBPA=PB, why is OPOP useful?
A.It is common to both right triangles OAPOAP and OBPOBP.
B.It is the tangent length.
C.It is always equal to PAPA.
D.It is a chord of the circle.
Question 22mcq
Which proof conclusion is properly justified?
A.Since OA=OBOA=OB, OP=OPOP=OP, and right angles at AA and BB, △OAP≅△OBP\triangle OAP\cong\triangle OBP, so PA=PBPA=PB.
B.Since the diagram looks symmetric, PA=PBPA=PB.
C.Since PP is outside, all segments from PP are equal.
D.Since OAOA is radius, PAPA is radius.
Question 23mcq
Which statement is unsafe without proof?
A.Line ll is tangent because it appears to touch the circle in the sketch.
B.A tangent touches a circle at one point.
C.Radius to point of contact is perpendicular to tangent.
D.Tangents from same external point are equal.
Question 24mcq
Assertion: A radius drawn to the point of contact of a tangent is perpendicular to the tangent. Reason: A tangent touches the circle at exactly one point. Which is correct?
A.Both are true, but the reason alone does not fully explain the perpendicularity.
B.Both are true and the reason fully proves it.
C.Assertion is true but reason is false.
D.Assertion is false but reason is true.