Question 1
How many tangents can a circle have?
Steps only
Every point of a circle has one tangent, and a circle has infinitely many points.
Answer
Infinitely many tangents.
CLASS 10 · MATHEMATICS · CHAPTER 10
Circles
Use the perpendicular-radius and equal-tangent theorems to calculate lengths and write clear geometric proofs.
20 worked answersAll 2 exercise sets
What you will learn
- Recognise tangents, secants and points of contact.
- Use the radius–tangent right angle.
- Prove results with equal tangents from one external point.
20 worked answers · short steps first, explanations when you need them
Exercise 10.1
Textbook pages 147–148
Question 2(i)
Complete: A tangent to a circle intersects it in ___ point(s).
Steps only
A tangent touches the circle at exactly one point.
Answer
One.
Question 2(ii)
Complete: A line intersecting a circle at two points is called a ___.
Steps only
A two-point intersecting line is a secant.
Answer
Secant.
Question 2(iii)
Complete: A circle can have at most ___ parallel tangents.
Steps only
One tangent can touch each of two opposite sides of the circle.
Answer
Two.
Question 2(iv)
Complete: The common point of a tangent and a circle is called the ___.
Steps only
It is where the tangent touches the circle.
Answer
Point of contact.
Question 3
A tangent PQ touches a circle of radius 5 cm at P. If OQ = 12 cm, find PQ.
Steps only
OP ⟂ PQ, so PQ² = OQ² − OP²
PQ = √(12² − 5²) = √119
Answer
√119 cm; option (D).
Question 4
Draw a circle and two lines parallel to a given line, one tangent and one secant.
Steps only
Draw the first parallel line at one radius from the centre and the second closer than one radius.
O(0, 0) · T(-3, 2) · U(3, 2) · S(-3, 1) · V(3, 1)
Answer
The first line touches once; the second crosses twice.
Exercise 10.2
Textbook pages 151–153
Question 1
From Q, a tangent is 24 cm and Q is 25 cm from the centre. Find the radius.
Steps only
r² + 24² = 25²
r² = 49, so r = 7
Answer
7 cm; option (A).
Question 2
Tangents TP and TQ touch a circle with centre O. If ∠POQ = 110°, find ∠PTQ.
Steps only
∠OPT = ∠OQT = 90°
∠PTQ = 360° − 90° − 90° − 110° = 70°
Answer
70°; option (B).
Question 3
Tangents PA and PB make an 80° angle at P. Find ∠POA.
Steps only
OP bisects ∠APB, so ∠APO = 40°
∠OAP = 90°, hence ∠POA = 50°
Answer
50°; option (A).
Question 4
Prove that tangents at the endpoints of a diameter are parallel.
Steps only
Both tangents are perpendicular to the same diameter.
Lines perpendicular to the same line are parallel.
Answer
The two tangents are parallel.
Question 5
Prove that the perpendicular at the point of contact to a tangent passes through the circle's centre.
Steps only
The radius through the contact point is perpendicular to the tangent.
Only one perpendicular can pass through that point.
Answer
The perpendicular is the radius line and therefore passes through the centre.
Question 6
A point A is 5 cm from a circle's centre and its tangent is 4 cm long. Find the radius.
Steps only
r² + 4² = 5²
r = √9 = 3
Answer
3 cm.
Question 7
Concentric circles have radii 5 cm and 3 cm. Find the chord of the larger circle tangent to the smaller circle.
Steps only
Half-chord = √(5² − 3²) = 4
Chord = 2 × 4 = 8
Answer
8 cm.
Question 8
ABCD circumscribes a circle. Prove AB + CD = AD + BC.
Steps only
Tangents from each vertex to its two contact points are equal.
AB + CD = AD + BC after pairing these equal lengths.
Answer
AB + CD = AD + BC.
Question 9
Two parallel tangents are met by a third tangent at A and B. Prove ∠AOB = 90°.
Steps only
OA and OB bisect the two angles made by the third tangent.
Those angles are supplementary, so their halves add to 90°.
Answer
∠AOB = 90°.
Question 10
Prove that the angle between two tangents from an external point is supplementary to the central angle between their contact radii.
Steps only
∠OAP = ∠OBP = 90°
∠APB + ∠AOB = 360° − 180° = 180°
Answer
The two angles are supplementary.
Question 11
Prove that a parallelogram circumscribing a circle is a rhombus.
Steps only
AB + CD = AD + BC
AB = CD and AD = BC, so 2AB = 2AD
Answer
All four sides are equal; it is a rhombus.
Question 12
A triangle circumscribes a circle of radius 4 cm. The contact point on BC gives BD = 8 cm and DC = 6 cm. Find AB and AC.
Steps only
Let equal tangents from A have length x. Then AB = x + 8, AC = x + 6 and BC = 14.
Semiperimeter s = x + 14; area = rs = 4(x + 14)
Heron: area² = (x + 14)·x·8·6. Solving gives x = 7
Answer
AB = 15 cm and AC = 13 cm.
Question 13
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre.
Steps only
∠AOB = 180° − (∠A + ∠B)/2
∠COD = 180° − (∠C + ∠D)/2
Their sum = 360° − 1/2(360°) = 180°
Answer
The opposite-side central angles are supplementary.