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Circles Practice 2: Lengths and Equal Tangents

Section A: Right Triangle Lengths

Question 1mcq
From external point PP, tangent PTPT touches a circle at TT. If OP=13OP=13 cm and radius OT=5OT=5 cm, find PTPT.
A.88 cm
B.1010 cm
C.1212 cm
D.1414 cm
Question 2mcq
A tangent from PP has length 2424 cm and the radius to the point of contact is 77 cm. Find OPOP.
A.2525 cm
B.2626 cm
C.3030 cm
D.3131 cm
Question 3mcq
If two tangents PAPA and PBPB are drawn from the same external point PP to a circle, what is true?
A.PA=PBPA=PB
B.PA⊥PBPA\perp PB
C.PAPA is a diameter
D.PBPB passes through the centre
Question 4mcq
Which theorem explains why two tangent segments from the same external point are equal?
A.Equal tangents theorem
B.Pythagoras theorem only
C.BPT
D.AA similarity
Question 5mcq
In proving equal tangents from external point PP, which triangle pair is usually shown congruent?
A.△OAP\triangle OAP and △OBP\triangle OBP
B.△AOB\triangle AOB and △APB\triangle APB
C.△OAB\triangle OAB and △PAB\triangle PAB
D.△OPA\triangle OPA and △APB\triangle APB

Section B: Equal Tangents and Perimeters

Question 6mcq
In the equal tangents proof, why are ∠OAP\angle OAP and ∠OBP\angle OBP right angles?
A.Radius is perpendicular to tangent at the point of contact.
B.Tangents are always parallel.
C.All angles in a circle are right angles.
D.OPOP bisects ABAB always.
Question 7mcq
Tangents PAPA and PBPB are drawn from point PP. If PA=11PA=11 cm, find PBPB.
A.5.55.5 cm
B.1010 cm
C.1111 cm
D.2222 cm
Question 8mcq
From point PP, tangents to a circle are PA=3x+2PA=3x+2 and PB=5x−6PB=5x-6. Find xx.
A.22
B.33
C.44
D.55
Question 9mcq
A quadrilateral is formed by tangents from two external points. From point PP, tangent lengths are 77 cm each; from point QQ, tangent lengths are 55 cm each. What is the perimeter?
A.1212 cm
B.2020 cm
C.2424 cm
D.2828 cm
Question 10mcq
In a tangential quadrilateral, two tangent segments from one vertex are x+3x+3 and 2x−12x-1. Find each segment length.
A.33 cm
B.55 cm
C.77 cm
D.99 cm