1
A line that touches a circle at exactly one point is —
💡 Hint:
One common point.
Explanation:
A tangent.
Class 10 · Maths · Chapter 10 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Circles
Circles
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- 1 The tangent, the limit of a secant
- 2 The chord shrinks to a point
- 3 The radius is perpendicular to the tangent
- 4 None inside, one on the circle, two outside
- 5 The two tangent segments from outside are equal
- 6 Angles, and the chord that touches the inner circle
- Chapter winner — every lesson at mastery ★
🎬 Shorts
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📄 PDF download — notes + answer key
The tangent, the limit of a secant
स्पर्श रेखा, छेदक की सीमा · NCERT 10.2 · Activity 1 · one point of contact
Touching at one pointNotes
A line that touches a circle at one point is a tangent. That common point is called the point of contact.
In Activity 1 a wire turns about P. When the second cut arrives at P, the line has become a tangent. A secant cuts at two points. There is one tangent at every point of the circle, so there are infinitely many tangents. There are at most two parallel tangents, one on each side.
Turn the wire until the second point becomes PActivity
A straight wire is fixed at P on a circular wire. As it turns, it cuts the circle at P and at some Q. Q moves slowly toward P. When the two points become one, the line no longer cuts, it touches. That is the mark of Activity 1.
Worked exampleExample
Question: At how many points must a line cut a circle to be a secant, and at how many to be a tangent?
Formula: Two points = secant. One point = tangent.
Substitution: Count the cutting points. If there are none, the line misses the circle.
Answer: A secant at two points. A tangent at exactly one point. The point of contact is that one point.
10-second revision
- A tangent touches at one point
- A secant cuts at two points
- Infinitely many tangents, at most two parallel
Board tip · BSEBBoard tip
In a BSEB answer write the name point of contact on its own. Only “it touches” is thin.
Board tip · CBSEBoard tip
On CBSE do not write “a circle has one tangent”. There are infinitely many.
Check your understandingall correct = mastery ★
2
A circle has infinitely many tangents.
Explanation:
True — one at every point.
3
The common point of a tangent and the circle is called the ______.
💡 Hint:
Where it touches.
Explanation:
The point of contact.
4
A line that cuts a circle at two points is —
💡 Hint:
Two cuts.
Explanation:
A secant.
5
A circle can have three parallel tangents.
Explanation:
False — at most two.
6
In Activity 1, how does turning the wire make a tangent?
Model answer (board format)
The second cutting point moves toward P and finally becomes P. The line then touches at one point, so it is a tangent.
Check your answer:
The chord shrinks to a point
जीवा सिकुड़कर बिंदु बनती है · Activity 2 · lines parallel to a secant
A tangent appears when the chord length becomes zeroNotes
In Activity 2 draw lines parallel to a secant. The chord gets shorter and on one side its length becomes zero. That line is the tangent.
This gives the same fact as Activity 1: a tangent is the secant whose chord has both ends at one point. If the line is completely outside the circle, there is no cut, and it is not a tangent.
Draw the parallel lines on both sidesActivity
Draw a secant PQ. Draw lines parallel to it above and below. The middle chord is long. Toward the edge the chord gets shorter. Call the line where the two points meet a tangent. You will find one tangent on the other side too. So a parallel pair stops at two.
Worked exampleExample
Question: Among lines parallel to a secant the chord is measured as 8 cm, then 3 cm, then 0 cm. What is the last line?
Formula: If the chord length is 0, both ends are one point.
Substitution: 8 cm and 3 cm still give two points, so those lines are secants. At 0 cm one point remains.
Answer: The last line is a tangent. The unit cm belonged to the chord. On the tangent the chord length is 0 cm.
10-second revision
- A chord of 0 is the tangent
- Activity 2 repeats Activity 1
- An outside line with no cut is not a tangent
Board tip · BSEBBoard tip
In a BSEB answer do not write 0 cm as “no line”. That line is the tangent.
Board tip · CBSEBoard tip
On CBSE do not reverse Activity 1 and Activity 2. Both give the same conclusion.
Check your understandingall correct = mastery ★
1
If the chord length is 0 cm, the line is —
💡 Hint:
Two ends, one point.
Explanation:
A tangent.
2
Activity 2 draws lines parallel to a secant.
Explanation:
True.
3
There are at most ______ parallel tangents.
💡 Hint:
One on each side.
Explanation:
2.
4
Of two parallel lines with chords 8 cm and 0 cm, which is the secant and which is the tangent?
Model answer (board format)
The 8 cm line is a secant, because there are two points. The 0 cm line is a tangent.
Check your answer:
The radius is perpendicular to the tangent
त्रिज्या स्पर्श रेखा पर लंब है · Theorem 10.1 · Exercise 10.1
The shortest distance is the perpendicularNotes
Among the points of the tangent, the point of contact P is the nearest to the centre O. So OP is perpendicular to the tangent. That is theorem 10.1.
If Q is outside and OQ is known, the tangent length PQ = √(OQ² − r²). If the radius is 5 cm and OQ = 13 cm, then PQ = 12 cm. If OQ = 12 cm and r = 5 cm, then PQ = √119 cm, not 13 cm. Question 3 of exercise 10.1 is this caution.
Read Pythagoras backwardsActivity
The right angle is at P. The hypotenuse is OQ. PQ² = OQ² − OP². 13² − 5² = 169 − 25 = 144, and the root is 12. 12² − 5² = 144 − 25 = 119, and the root is √119. A negative length is not taken.
Worked exampleExample
Question: The radius is 5 cm. The distance from the centre to the outside point Q is 13 cm. Find the tangent length PQ.
Formula: PQ² = OQ² − r².
Substitution: PQ² = 13² − 5² = 169 − 25 = 144.
Answer: PQ = 12 cm. Check: 5-12-13.
10-second revision
- OP is perpendicular to the tangent
- PQ = √(OQ² − r²)
- When OQ = 12 and r = 5, the length is √119 cm
Board tip · BSEBBoard tip
In a BSEB answer show the line 169 − 25 = 144. Jumping to 12 cm is thin.
Board tip · CBSEBoard tip
On CBSE do not turn √119 into 13. 13 appears when OQ = 13.
Check your understandingall correct = mastery ★
1
r = 5 cm, OQ = 13 cm. PQ is —
💡 Hint:
169 − 25.
Explanation:
12 cm.
2
If r = 5 cm and OQ = 12 cm, then PQ = 13 cm.
Explanation:
False — PQ = √(144 − 25) = √119 cm.
3
In theorem 10.1 the angle between OP and the tangent is ______°.
💡 Hint:
Perpendicular.
Explanation:
90°.
4
The perpendicular to the tangent at the point of contact —
💡 Hint:
The converse use of theorem 10.1.
Explanation:
That perpendicular passes through the centre.
5
r = 9 cm and OQ = 15 cm. Find the tangent length.
Model answer (board format)
PQ² = 225 − 81 = 144. PQ = 12 cm.
Check your answer:
None inside, one on the circle, two outside
अंदर शून्य, ऊपर एक, बाहर दो · Activity 3 · the count before exercise 10.2
The place decides the countNotes
Activity 3 checks three places. Inside 0, on the circle 1, outside 2.
From an inside point every line cuts the circle twice, so no tangent forms. From a point on the circle there is exactly the one tangent of theorem 10.1. From outside, two tangents form and their points of contact are different. Exercise 10.2 asks for the angles and the lengths of these two lines.
Write the place of the point firstActivity
Draw a circle on paper. Put the point inside and turn a line: always two cuts. Put the point on the circle: one tangent. Put the point outside: exactly two lines that touch. The count 0, 1, 2 is the thing to remember.
| Point | Tangents |
|---|---|
| Inside | 0 |
| On the circle | 1 |
| Outside | 2 |
Worked exampleExample
Question: A point is 3 cm from the centre and the radius is 5 cm. How many tangents are there from that point?
Formula: Compare the distance from the centre with the radius.
Substitution: 3 cm < 5 cm, so the point is inside.
Answer: 0 tangents. If the distance were 5 cm there would be 1, and if it were 8 cm there would be 2.
10-second revision
- Inside: 0
- On the circle: 1
- Outside: 2
Board tip · BSEBBoard tip
In a BSEB answer write the line that compares the distance and the radius, then the count.
Board tip · CBSEBoard tip
On CBSE do not write “infinitely many tangents from outside”. From one point there are two.
Check your understandingall correct = mastery ★
1
From a point inside the circle the tangents are —
💡 Hint:
Every line cuts twice.
Explanation:
0.
2
There is exactly one tangent from a point on the circle.
Explanation:
True.
3
From one outside point there are ______ tangents.
💡 Hint:
Activity 3.
Explanation:
2.
4
Radius 5 cm and distance from the centre 5 cm. Tangents —
💡 Hint:
The point is on the circle.
Explanation:
1.
5
If the centre is 8 cm away and the radius is 5 cm, there are no tangents.
Explanation:
False — the point is outside, so 2.
6
The radius is 6 cm. Points are 6 cm and 10 cm from the centre. Write the two counts.
Model answer (board format)
At 6 cm the point is on the circle, so 1 tangent. At 10 cm the point is outside, so 2 tangents.
Check your answer:
The two tangent segments from outside are equal
बाहर से दोनों स्पर्श खंड बराबर · Theorem 10.2 · Exercise 10.2 · lengths
One outside point, two equal lengthsNotes
From an outside point T the points of contact are P and Q. TP = TQ. That is theorem 10.2.
The length is Pythagoras again. If the tangent segment is 8 cm and the distance from the centre to T is 10 cm, the radius is √(100 − 64) = 6 cm. The first question of exercise 10.2 is the larger form of this triple: 7 cm, 24 cm, 25 cm. In a circumscribed quadrilateral the equal tangent segments give AB + CD = AD + BC.
Keep the length that is asked as the unknownActivity
OQ is the hypotenuse. The tangent length and the radius are the legs. If two of the three are given, find the third. With a 24 cm tangent and a 25 cm centre-distance, r = √(625 − 576) = √49 = 7 cm. From 8 and 10, r = 6 cm.
Worked exampleExample
Question: The tangent segment from an outside point is 8 cm and the distance from the centre to that point is 10 cm. Find the radius.
Formula: r² = OQ² − (tangent)².
Substitution: r² = 10² − 8² = 100 − 64 = 36.
Answer: r = 6 cm. The other tangent segment is also 8 cm.
10-second revision
- TP = TQ
- r = √(OQ² − tangent²)
- AB + CD = AD + BC
Board tip · BSEBBoard tip
In a BSEB answer write both the name theorem 10.2 and TP = TQ.
Board tip · CBSEBoard tip
On CBSE, with 24 cm and 25 cm, the radius is 7 cm, not 24.5 cm.
Check your understandingall correct = mastery ★
1
Tangent 8 cm, distance from the centre 10 cm. The radius is —
💡 Hint:
100 − 64.
Explanation:
6 cm.
2
The two tangent segments from one outside point are equal.
Explanation:
True — theorem 10.2.
3
If the tangent is 24 cm and the centre-distance is 25 cm, the radius is ______ cm.
💡 Hint:
625 − 576.
Explanation:
7 cm.
4
In circumscribed quadrilateral ABCD, AB = 6 cm, BC = 7 cm, CD = 8 cm. Find AD.
Model answer (board format)
AB + CD = AD + BC. 6 + 8 = AD + 7. AD = 7 cm.
Check your answer:
Angles, and the chord that touches the inner circle
कोण और छोटी वृत्त को छूती जीवा · Exercise 10.2 · 70°, 50° and the chord
Two right angles sit in the quadrilateralNotes
OP and OQ are perpendicular to the tangents. Angle PTQ = 180° − angle POQ. If angle POQ = 110°, then angle PTQ = 70°.
If the outside angle is 80°, OP bisects it, so 40°. In triangle OAP, after 90° + 40°, angle POA = 50°. For concentric circles of radii 10 cm and 8 cm, half of the touching chord is √(100 − 64) = 6 cm, and the whole chord is 12 cm.
Subtract two right angles from 360°Activity
In quadrilateral OPTQ, 90° + 90° + 110° = 290°. The angle left is 70°. In the question with an outside angle of 80°, half is 40° and 180° − 90° − 40° = 50°. In the chord question, find half the chord and double it.
Worked exampleExample
Question: Two concentric circles have radii 10 cm and 8 cm. Find the chord of the larger circle that touches the smaller one.
Formula: Half the chord = √(R² − r²).
Substitution: √(10² − 8²) = √(100 − 64) = √36 = 6.
Answer: The whole chord = 12 cm.
10-second revision
- Angle PTQ = 180° − angle POQ
- An outside 80° gives angle POA = 50°
- Chord = 2√(R² − r²)
Board tip · BSEBBoard tip
In a BSEB answer show the line 90° + 90°, then the angle that remains.
Board tip · CBSEBoard tip
On CBSE do not leave half the chord as the whole answer. 6 cm is half, 12 cm is the whole.
Check your understandingall correct = mastery ★
1
If angle POQ = 110°, angle PTQ is —
💡 Hint:
180 − 110.
Explanation:
70°.
2
If two tangents meet at 80°, angle POA = 50°.
Explanation:
True — 180 − 90 − 40.
3
If the radii are 10 cm and 8 cm, the whole touching chord is ______ cm.
💡 Hint:
2 × 6.
Explanation:
12 cm.
4
In the quadrilateral with angle POQ = 110°, the angles at the points of contact are —
💡 Hint:
Theorem 10.1.
Explanation:
Both are 90°.
5
If angle POQ = 120°, find the outside angle PTQ. Write the reason in one line.
Model answer (board format)
There are 90° + 90° at the points of contact. PTQ = 180° − 120° = 60°.
Check your answer:
❓ Full question bank — with answers and explanations — 65 questions
Pick a type. The 30 lesson checks are separate — each lesson has as many as its topic needs. All correct earns mastery ★.
BSEB modelpractice · not annual60 tried · 0 correctMultiple choiceMCQ120 tried · 0 correctTrue / falseTrue / false80 tried · 0 correctFill in the blanksFill80 tried · 0 correctMatchMatch20 tried · 0 correctAssertion–reasonAssertion–reason50 tried · 0 correctBalancingBalancing40 tried · 0 correctIdentifyIdentify20 tried · 0 correctShort and longVSA, SA, LA120 tried · 0 correctCBSE-stylenot a PYQ60 tried · 0 correct
No question is marked as a verified past paper. The BSEB set is a model for practice. CBSE items are CBSE-style, not a copy of any year’s paper.
Multiple choice
0/12Pick one option. A wrong try brings a hint.
1
A line that touches a circle at one point is —
💡 Hint:
One point.
Explanation:
A tangent.
2
Parallel tangents are at most —
💡 Hint:
One on each side.
Explanation:
2.
3
r = 5 cm, OQ = 13 cm. Tangent length —
💡 Hint:
169 − 25.
Explanation:
12 cm.
4
r = 5 cm, OQ = 12 cm. Tangent length —
💡 Hint:
144 − 25.
Explanation:
√119 cm.
5
From one point outside the circle, the tangents are —
💡 Hint:
Activity 3.
Explanation:
2.
6
Tangent 24 cm, centre-distance 25 cm. Radius —
💡 Hint:
625 − 576.
Explanation:
7 cm.
7
Angle POQ = 110°. Angle PTQ —
💡 Hint:
180 − 110.
Explanation:
70°.
8
Outside angle 80°. Angle POA —
💡 Hint:
180 − 90 − 40.
Explanation:
50°.
9
Radii 10 cm and 8 cm. Whole chord —
💡 Hint:
2 × 6.
Explanation:
12 cm.
10
Tangent 8 cm, 10 cm from the centre. Radius —
💡 Hint:
100 − 64.
Explanation:
6 cm.
11
AB = 6, BC = 7, CD = 8, circumscribed. AD —
💡 Hint:
6 + 8 = AD + 7.
Explanation:
7 cm.
12
If the distance from the centre is less than the radius, the tangents are —
💡 Hint:
The point is inside.
Explanation:
0.
True or false
0/81
A secant cuts a circle at two points.
Explanation:
True.
2
The tangent is parallel to the radius.
Explanation:
False — it is perpendicular to the radius.
3
There is exactly one tangent from a point on the circle.
Explanation:
True.
4
TP and TQ from one outside point are equal.
Explanation:
True — theorem 10.2.
5
When OQ = 12 cm and r = 5 cm, the tangent length is 13 cm.
Explanation:
False — √119 cm.
6
In a circumscribed quadrilateral, AB + CD = AD + BC.
Explanation:
True.
7
A circle has only two tangents.
Explanation:
False — infinitely many. Parallel ones are at most two.
8
If angle POQ = 120°, then angle PTQ = 60°.
Explanation:
True — 180 − 120.
Fill in the blanks
0/81
A line that cuts at two points is a ______.
💡 Hint:
A secant.
Explanation:
Not a tangent.
2
The angle between OP and the tangent is ______°.
💡 Hint:
90.
Explanation:
Theorem 10.1.
3
r = 5, OQ = 13. PQ = ______ cm.
💡 Hint:
12 cm.
Explanation:
5-12-13.
4
The number of tangents from outside is ______.
💡 Hint:
2.
Explanation:
Activity 3.
5
With 24 cm and 25 cm the radius is ______ cm.
💡 Hint:
7 cm.
Explanation:
7-24-25.
6
If angle POQ = 110°, PTQ = ______°.
💡 Hint:
70.
Explanation:
180 − 110.
7
For radii 10 and 8 the whole chord is ______ cm.
💡 Hint:
12 cm.
Explanation:
2√36.
8
AB + CD = AD + ______.
💡 Hint:
BC.
Explanation:
A circumscribed quadrilateral.
Match
0/21
Match the name with its meaning.
Column B: A. TP = TQ · B. One point of contact · C. Two cuts · D. Radius perpendicular
1. Tangent
2. Secant
3. Theorem 10.1
4. Theorem 10.2
Explanation:
A tangent is one point, a secant is two cuts, 10.1 is the right angle, and 10.2 is equal lengths.
2
Match the activity or exercise with its job.
Column B: A. 70° when the centre has 110° · B. Turn a wire to one point · C. 0, 1, 2 · D. √(OQ² − r²)
1. Activity 1
2. Activity 3
3. The length in 10.1
4. The angle in 10.2
Explanation:
Activity 1 is the limit, Activity 3 is the count, 10.1 is the length, and 10.2 is the angle.
Assertion–reason
0/5Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): If r = 5 cm and OQ = 13 cm, the tangent length is 12 cm.
Reason (R): PQ² = OQ² − r² and 169 − 25 = 144.
Explanation:
Both are true and R is the reason.
2
Assertion (A): A circle has infinitely many tangents.
Reason (R): From one outside point there are exactly two tangents.
Explanation:
Both are true, but R does not explain the infinite count. It is infinite because there is one tangent at every point.
3
Assertion (A): If angle POQ = 110°, then angle PTQ = 70°.
Reason (R): The angles at the points of contact are 80°.
Explanation:
A is true. R is false, those angles are 90°.
4
Assertion (A): Two tangents can be drawn from a point inside the circle.
Reason (R): From inside, every line cuts the circle at two points.
Explanation:
A is false. R is true, so the count is 0.
5
Assertion (A): For radii 10 cm and 8 cm the touching chord is 12 cm.
Reason (R): Half the chord is √(100 − 64) = 6 cm.
Explanation:
Both are true and R is the half length, the whole is 12 cm.
Coefficient practice
0/4These chemistry lines are not a result of this maths chapter. They are only practice in filling coefficients. A blank means 1.
1
This is coefficient practice and not a result of this maths chapter. In the line where H2 + O2 makes water, Fill coefficients (blank = 1).
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
2
This is coefficient practice and not a result of this maths chapter. In the line where N2 + H2 makes ammonia, Fill coefficients (blank = 1).
N2 + H2 → NH3
Explanation: N2 + 3H2 → 2NH3
3
This is coefficient practice and not a result of this maths chapter. In the line where C + O2 makes carbon dioxide, Fill coefficients (blank = 1).
C + O2 → CO2
Explanation: C + O2 → CO2
4
This is coefficient practice and not a result of this maths chapter. In the burning line of CH4 + O2, Fill coefficients (blank = 1).
CH4 + O2 → CO2 + H2O
Explanation: CH4 + 2O2 → CO2 + 2H2O
Classify
0/21
Place each line as tangent, secant, or missing.
One point of contact
Two cuts
No common point
Chord length 0 cm
Explanation:
One point is a tangent. Two cuts are a secant. No point means the line misses. A 0 cm chord is a tangent.
2
Place each point with its tangent count.
Distance from the centre less than the radius
Distance equal to the radius
Distance greater than the radius
The greatest count of parallel tangents
Explanation:
Inside 0, on the circle 1, outside 2. Parallel tangents are also at most 2.
Very short answer
0/51
Define a tangent.
Model answer (board format)
A line that touches a circle at exactly one point is a tangent.
Check your answer:
2
Write theorem 10.1.
Model answer (board format)
The radius through the point of contact is perpendicular to the tangent.
Check your answer:
3
Write theorem 10.2.
Model answer (board format)
The segments of the tangents drawn from one outside point are equal.
Check your answer:
4
Write the counts for inside, on the circle and outside.
Model answer (board format)
0, 1 and 2.
Check your answer:
5
Write the side relation of a circumscribed quadrilateral.
Model answer (board format)
AB + CD = AD + BC.
Check your answer:
Short answer
0/41
r = 8 cm, OQ = 17 cm. Find the tangent length.
Model answer (board format)
PQ² = 289 − 64 = 225. PQ = 15 cm.
Check your answer:
2
The tangent segment is 12 cm and the radius is 5 cm. Find the distance from the centre to the outside point.
Model answer (board format)
OQ² = 144 + 25 = 169. OQ = 13 cm.
Check your answer:
3
If angle POQ = 100°, find angle PTQ.
Model answer (board format)
PTQ = 180° − 100° = 80°. There are 90° at the points of contact.
Check your answer:
4
The concentric radii are 13 cm and 5 cm. Find the chord of the larger circle that touches the smaller one.
Model answer (board format)
Half the chord is √(169 − 25) = √144 = 12 cm. The whole chord is 24 cm.
Check your answer:
Long answer
0/31
Find the tangent length for r = 5 cm and OQ = 12 cm. Then say why it is not 13 cm. If OQ = 13 cm on the same circle, what is the length?
Model answer (board format)
PQ² = 144 − 25 = 119. PQ = √119 cm. 13 cm would be the case when the hypotenuse of a 5-12-13 triangle is 13. When OQ = 13, PQ² = 169 − 25 = 144, so PQ = 12 cm.
Check your answer:
2
Two tangents meet at 70°. Find angle POA at the centre and say why OP bisects the angle.
Model answer (board format)
TP = TQ, OP is common and the radii are equal, so the triangles are congruent and OP bisects the 70° angle. Half is 35°. Angle POA = 180° − 90° − 35° = 55°.
Check your answer:
3
A wheel rolls on a road. Why is the road a tangent? What angle does the radius make with the road? If the radius is 14 cm and an outside point is 50 cm from the hub, find the tangent length.
Model answer (board format)
The road touches the wheel at one point, so it is a tangent. The radius makes 90°. The length is √(2500 − 196) = √2304 = 48 cm.
Check your answer:
BSEB model paper · practice
0/6This model set is for practice. It is not a question from any year’s annual examination.
1
In exercise 10.1, for r = 5 cm and OQ = 12 cm, the correct choice is —
💡 Hint:
144 − 25.
Explanation:
√119 cm.
2
From an outside point the tangents are —
💡 Hint:
Theorem 10.2.
Explanation:
2 and equal.
3
If angle POQ = 140°, PTQ = ______°.
💡 Hint:
40°.
Explanation:
180 − 140.
4
The tangent length is 9 cm and the radius is 12 cm. Find the distance from the centre to the outside point.
Model answer (board format)
OQ² = 81 + 144 = 225. OQ = 15 cm.
Check your answer:
5
Three sides of a circumscribed quadrilateral are 5 cm, 6 cm and 7 cm, in order AB, BC, CD. Find AD. Then say why this relation comes from the tangent segments.
Model answer (board format)
AB + CD = AD + BC. 5 + 7 = AD + 6. AD = 6 cm. The two tangent segments from one outside corner are equal, so the sums of the opposite pairs become equal.
Check your answer:
6
Assertion (A): There is one tangent from a point on the circle.
Reason (R): Two equal tangent segments also form from that point.
Explanation:
A is true. R is false. Two segments form from an outside point.
CBSE-style questions
0/6These are competency-based practice questions. They are not a copy of any year’s paper.
1
A student picks 13 cm for OQ = 12 cm and r = 5 cm. The correct length is —
💡 Hint:
The hypotenuse is 12, not 13.
Explanation:
√(144 − 25) = √119 cm. 13 cm is the case when OQ = 13.
2
A wheel touches the road. The road is —
💡 Hint:
One point of contact.
Explanation:
A tangent, and the radius is perpendicular to it.
3
Assertion (A): In a circumscribed quadrilateral with AB = 5 cm, BC = 6 cm and CD = 7 cm, AD = 6 cm.
Reason (R): AB + CD = AD + BC.
Explanation:
Both are true. 5 + 7 = AD + 6.
4
Assertion (A): A tangent can be drawn from inside the circle.
Reason (R): A line through an inside point cuts the circle at two points.
Explanation:
A is false. R is true.
5
Two tangents meet at 100°. Find angle POA at the centre.
Model answer (board format)
OP bisects the angle, so 50°. The right angle is at the point of contact. POA = 180 − 90 − 50 = 40°.
Check your answer:
6
Concentric circles have radii 25 cm and 7 cm. Find the chord of the larger one that touches the smaller one.
Model answer (board format)
Half the chord is √(625 − 49) = √576 = 24 cm. The whole chord is 48 cm.
Check your answer:
🏛️ Board exam corner — Bihar Board (BSEB)— CBSE
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🧠 What you learned + equation sheet
What you learned
- A tangent touches at one point. A secant cuts at two points.
- The radius is perpendicular to the tangent at the point of contact.
- 0 tangents from inside, 1 from a point on the circle, 2 from outside.
- The two tangent segments from one outside point are equal.
- Tangent length² + radius² = distance from the centre².
- In a circumscribed quadrilateral, AB + CD = AD + BC.
| What | Keep this |
|---|---|
| Tangent | one point of contact |
| Secant | two intersection points |
| Theorem 10.1 | radius ⊥ tangent |
| Count | 0 inside, 1 on, 2 outside |
| Theorem 10.2 | equal tangent segments |
| Circumscribed | AB + CD = AD + BC |
The notes are original writing. The textbook was used only for activity order and numbers. “Verified” will be used only on a question that has a source page.