Engineering Drawing · CITS · NSQF Level 5
Tangent to Circle
External · Internal · Tangential Curves — Complete Study Guide
Theory + Concepts
20 Conceptual Questions
50 Master MCQs
50 Rapid Fire MCQs
1
The Why & How — 80/20 Rule
The Theory
Focus on the 20% of concepts that appear in 80% of exam questions and practical problems.
🔵 What Is a Tangent to a Circle?
- A tangent is a straight line that just touches a circle at exactly one point
- It does NOT cut through or pass inside the circle, even when extended infinitely
- This unique single-touch point is called the Point of Tangency
- The golden rule: The line joining the centre to the point of tangency is always perpendicular (90°) to the tangent
- This is the key geometric property that drives ALL tangent constructions
- Formula: OP ⊥ Tangent where O = centre, P = point of tangency
- A Normal to a circle at point P is the line OP itself — it is perpendicular to the tangent
🟠 Type 1 — Tangent at a Point on the Circle (Exercise 1.1)
- Given: Circle of known diameter, point P already on its circumference
- Construction steps:
- Draw the circle and mark the given point P on its edge
- Join the centre O to point P (this is the radius OP)
- Draw a line RS perpendicular to OP through P
- Line RS is the required tangent at P
- Core principle used: Tangent ⊥ Radius at point of tangency
🔴 Type 2 — External Tangent to Two Circles (Exercise 1.7)
External- What it is: A common tangent that does NOT cross between the two circles — it lies outside both
- Both circles lie on the same side of the tangent line
- Key construction trick:
- On the larger circle (D1), draw a concentric auxiliary circle of diameter = D1 − D2 (difference of diameters)
- Draw a tangent from the centre of the smaller circle to this auxiliary circle → This gives the direction of the external tangent
- Parallel lines from both centres define where the tangent meets each circle (P1, P2)
- Two external tangents T1 and T2 can be drawn (one on each side)
- Formula shortcut: Aux. circle dia = D₁ − D₂
🟢 Type 3 — Internal Tangent to Two Equal Circles (Exercise 1.8)
Internal- What it is: A common tangent that passes between the two circles — it crosses the line joining both centres
- The two circles lie on opposite sides of the tangent line
- Also called transverse common tangent
- Key construction method (equal circles):
- Find midpoint A of the line C1C2 joining the two centres
- Bisect C1A to get point B
- Draw arc from B (radius = C1B) → gives points P1 and P2 on the large semicircle
- Draw arc from A (radius = AP1) → gives points P3 and P4
- Join P1P4 and P2P3 to get the two internal tangents T1 and T2
- The crossing point of internal tangents always lies at the midpoint of C1C2
🟣 Type 4 — Tangential Curves (Exercise 1.9) & Three Mutually Tangent Circles (Exercise 1.6)
Tangential- Tangential arc / curve: An arc drawn to smoothly connect (be tangent to) two existing circles
- The connecting arc touches each circle at exactly one point with no sharp corner
- To find the centre of a tangential arc of radius R, draw an arc from each circle's centre with radius = (R + that circle's radius)
- The intersection of these two arcs is the centre of the required tangential curve
- R_arc from C₁ = R_tangential + R₁ R_arc from C₂ = R_tangential + R₂
- Three mutually tangent circles (triangle method):
- Join the three centres A, B, C to form a triangle
- Bisect any two angles of the triangle — the bisectors meet the opposite sides at F and G
- AF, BF (or BG), and CG are the radii of the three circles respectively
💡 Analogy — for External vs. Internal Tangents
Imagine two coins lying flat on a table. A ruler laid across the top of both coins (without going between them) is like an external tangent — it skims both circles from the outside. Now imagine the same ruler passed between the two coins, touching one from the left and the other from the right — that crossing line is an internal tangent.
📋 Quick-Reference Summary
- Tangent at a point: Draw radius to P → draw perpendicular at P → done
- External tangent (unequal circles): Auxiliary circle of dia (D1−D2) on larger circle → tangent from smaller circle's centre
- Internal tangent (equal circles): Midpoint method → bisect C1A → arc constructions → join crossing points
- Tangential curve: Sum of radii method → centre found by arc intersection → draw connecting arc
- 3 mutually tangential circles: Triangle on centres → angle bisectors → use bisector segments as radii
- Universal rule (never forget): Radius ⊥ Tangent at point of tangency = 90°
2
Hands-On Exercises
Practical
🛠️ Practical Exercises
Practical content — step-by-step construction procedures, observation tables, and figures — will be added here.
Follow your instructor's guidance for Exercises 1.1 through 1.9.
3
Deep Understanding
Conceptual Questions
These 20 short-answer questions test your logic, not just memory. Attempt each before looking anything up.
Q1
How does the angle between a radius and a tangent change if the tangent is not drawn at the exact point of tangency but slightly away from it?
Q2
What would happen to the external tangent construction if both circles had exactly the same diameter? Would the auxiliary circle still work?
Q3
How does increasing the centre distance between two circles affect the angle at which external tangents meet (or appear to meet) if extended?
Q4
Why must the auxiliary circle drawn on the larger circle have a diameter equal to the difference of the two circles' diameters during external tangent construction?
Q5
What would happen if you attempted to draw an internal tangent to two circles whose circumferences overlap or intersect each other?
Q6
How does the position of the midpoint of C1C2 relate to the crossing point of internal tangents? What geometric principle does this demonstrate?
Q7
If you increase the radius of the tangential arc connecting two circles, how does the smoothness and curvature of the connecting curve change?
Q8
In the three mutually tangential circles construction, why are the angle bisectors of the triangle used to find the radii rather than the altitudes or medians?
Q9
How would the tangential arc centre location change if the connecting arc needed to be internally tangent to one circle and externally tangent to the other?
Q10
What is the relationship between the normal to a circle and the tangent to that circle at the same point? How does this help in verifying a construction?
Q11
If two circles are concentric (same centre), can external or internal common tangents be drawn between them? Explain why or why not.
Q12
How does the angle bisector method for three mutually tangential circles change if one of the three centres is collinear with the other two?
Q13
What would happen geometrically if, during internal tangent construction, the two circles were so close that the midpoint A coincides with one of the centres?
Q14
How does the formula "arc radius from C1 = R_tangential + R1" ensure that the tangential arc will be externally tangent (not intersecting) to circle 1?
Q15
In practical engineering drawing, why is a tangent preferred over an intersecting arc when joining two curved features on a component?
Q16
What would change in the external tangent construction if the line from C2 touched the auxiliary circle at a point other than P (i.e., the tangent condition was not satisfied)?
Q17
How does drawing a perpendicular to the line OP at point P guarantee the tangent, rather than simply being a coincidence of construction?
Q18
If the centre distance C1C2 is less than the sum of the two radii, what type of relationship do the circles have, and can common tangents still be drawn?
Q19
What visual or measurement check can you perform after completing a tangent construction to verify that your tangent is correct?
Q20
How does the concept of a tangent to a circle extend to real-world applications in cam design, pulley belt layout, and gear tooth profiles?
4
Exam Standard — With Answers & Explanations
Master MCQs
50 high-quality MCQs. Distractor options are carefully chosen to be plausible but wrong. Read every explanation — that's where the learning happens.
Q1. A tangent to a circle is defined as a line that:
Answer: B — A tangent touches a circle at precisely one point (the point of tangency) and does not enter the circle. A line through the centre is a diameter-extension, and a line cutting the circle at two points is a secant.
Q2. The angle between a tangent and the radius drawn to the point of tangency is always:
Answer: C — This is the fundamental theorem of tangents: the radius to the point of tangency is always perpendicular (90°) to the tangent. This principle underlies every tangent construction.
Q3. The point where a tangent touches a circle is called the:
Answer: B — The specific point where the tangent line just touches the circle is called the "point of tangency." Vertex and node are terms used in different geometric contexts.
Q4. To draw a tangent to a circle at a given point P on its circumference, the first step is to:
Answer: B — The construction begins by drawing the radius OP. The tangent is then drawn perpendicular to this radius at point P. Without the radius, the perpendicular direction cannot be established.
Q5. In Exercise 1.7 (external tangent construction), an auxiliary concentric circle is drawn on the larger circle. Its diameter equals:
Answer: B — The auxiliary circle has diameter = D1 − D2 (40 − 30 = 10 mm in the example). This reduces the problem geometrically, offsetting by the radius of the smaller circle.
Q6. In the external tangent construction, after drawing the line from C2 tangent to the auxiliary circle at P, you then:
Answer: B — The extension of C1P meets circle D1 at P1. Then C2P2 is drawn parallel to C1P1. Joining P1 and P2 gives the external tangent T1.
Q7. Two external common tangents can be drawn to two circles when:
Answer: C — External common tangents require the circles to be fully separate (not overlapping or nested).
Q8. An internal tangent to two circles is one that:
Answer: A — An internal (transverse) common tangent crosses the space between the circles and intersects the line C1C2 at the midpoint.
Q9. In Exercise 1.8 (internal tangents, equal circles, centre distance 60 mm), point A is:
Answer: B — A is defined as the midpoint of C1C2. For equal circles, the internal tangents must cross exactly at this midpoint (by symmetry). Point B is then found by bisecting C1A.
Q10. In Exercise 1.8, after marking B (midpoint of C1A), the next step is to draw an arc with:
Answer: B — Using B as centre and C1B as radius, the arc intersects C1C2 at points P1 and P2. These become the basis for locating the tangent contact points P3 and P4.
Q11. In Exercise 1.9 (tangential curves), circles of diameter 30 and 20 mm have centres 60 mm apart. The arc from C1 to find the connecting arc's centre has radius:
Answer: C — R1 = 15 mm (half of 30), connecting arc radius = 20 mm. Arc from C1 = R1 + R_connecting = 15 + 20 = 35 mm.
Q12. In Exercise 1.9, the arc from C2 to help locate the tangential curve centre has radius:
Answer: B — R2 = 10 mm (half of 20), connecting arc radius = 20 mm. Arc from C2 = 10 + 20 = 30 mm.
Q13. For the three mutually tangential circles (Exercise 1.6), after forming triangle ABC, which lines are used to find the radii?
Answer: C — Bisecting any two angles gives bisectors that meet opposite sides at F and G. These segments (AF, BF/BG, CG) become the radii of the three tangential circles.
Q14. How many external common tangents can generally be drawn to two circles that do not overlap?
Answer: C — Two circles fully external to each other have exactly 2 external common tangents (one on each side).
Q15. The normal to a circle at any point P is:
Answer: C — The normal to a circle at any point P is the line that passes through the centre O and the point P (i.e., the radius line). It is perpendicular to the tangent at P.
Q16. In the external tangent construction for circles of dia 40 and 30 mm (centre distance 60 mm), the auxiliary circle on D1 has a diameter of:
Answer: A — Auxiliary diameter = D1 − D2 = 40 − 30 = 10 mm.
Q17. A line that intersects a circle at exactly two points is called a:
Answer: C — A secant crosses a circle at two points. A tangent touches at one. A chord is a line segment (not extended) between two circle points.
Q18. In Exercise 1.8, the circles have the same diameter of 40 mm and centres 60 mm apart. How many internal tangents can be drawn?
Answer: B — Two internal tangents (T1: P1P4 and T2: P2P3) are drawn. They cross at the midpoint of C1C2.
Q19. In the tangential curve exercise, C3 and C4 are found as centres. With what radius are the final tangential arcs drawn?
Answer: B — From C3 and C4 as centres, arcs of radius 20 mm are drawn — the given connecting arc radius.
Q20. A line drawn perpendicular to a tangent at the point of tangency will always pass through:
Answer: C — A perpendicular drawn to the tangent at P is the radius (or normal), and it always passes through the centre O.
Q21. Which of the following best describes "concentric circles"?
Answer: B — Concentric circles share the same centre but have different radii. Ball bearings are a practical example.
Q22. Eccentric circles are defined as:
Answer: C — Eccentric circles are nested inside another circle but their centres do not coincide.
Q23. The angle C1PP (where P is the tangent point on the auxiliary circle) in the external tangent construction is:
Answer: C — The line from C2 is tangent to the auxiliary circle at P, meaning it is perpendicular to the radius C1P at that point (tangent–radius rule).
Q24. Two internally tangent circles (one inside the other, touching at one point) can have how many common tangents?
Answer: A — When one circle is internally tangent to another, only 1 common tangent exists at that single touch point.
Q25. In Exercise 1.1, if the circle has a diameter of 70 mm, the tangent is drawn perpendicular to which line?
Answer: B — "Draw a line RS perpendicular to PO through P" — OP is the radius.
Q26. The connecting arc in a tangential curve construction is designed so that it touches each circle at exactly one point with no abrupt change in direction. This is called:
Answer: B — Tangential means smooth contact at one point. The arc merges with each circle seamlessly (no kink or corner).
Q27. What is the total number of common tangents possible for two separate non-overlapping circles?
Answer: C — For two fully separate circles: 2 external + 2 internal = 4 common tangents total.
Q28. In a tangential arc construction, if the connecting arc of radius R is to be externally tangent to a circle of radius r1, the locus of the arc's centre from C1 has radius:
Answer: C — For external tangency, two circle centres are separated by the sum of their radii (R + r1).
Q29. For the three mutually tangential circles (Exercise 1.6), the radius of the circle centred at A is:
Answer: B — The angle bisector from A meets opposite side BC at F. AF is the radius for circle A.
Q30. In the internal tangent construction, lines P1P4 and P2P3 intersect at:
Answer: C — Internal tangents always cross at the midpoint of C1C2 (point A) for equal circles.
Q31. Which of the following is a real-life application of tangent concepts in engineering drawing?
Answer: B — Belt drives require external/internal tangent lines. Cam profiles use tangential curves to ensure smooth motion.
Q32. When drawing an arc tangent to two lines that form an acute angle, the centre of the arc is found by:
Answer: B — Parallel lines at radius distance from each original line intersect at the arc centre O, equidistant from both — guaranteeing tangency.
Q33. The minimum number of common tangents for two circles touching each other externally (at one outside point) is:
Answer: B — When two circles touch externally: 2 external + 1 common tangent at the touch point = 3 total.
Q34. The construction of tangential curves uses two centres C3 and C4. Why are two centres found?
Answer: A — Two intersection points C3 and C4 (above and below centreline) give two symmetric tangential arcs, one on each side.
Q35. In Exercise 1.6 (three mutually tangential circles), radius of circle at B equals:
Answer: C — BF and BG are both valid radii for circle B. The textbook states "BF or BG as radius."
Q36. When two circles have the same diameter, can external tangents still be drawn using the standard auxiliary-circle method?
Answer: A — When D1 = D2, auxiliary circle diameter = 0. For equal circles, external tangents are simply parallel lines at perpendicular distance equal to the radius.
Q37. A tangential curve ensures smooth blending between two circles primarily because:
Answer: B — Smooth blending occurs because at each contact point, the connecting arc and the circle share a common tangent line — their radii are collinear at the contact point.
Q38. Which of the following statements about external and internal tangents is INCORRECT?
Answer: C — Internal and external tangents are generally NOT parallel. They are distinct lines that cross each other.
Q39. If the centre distance between two circles equals the difference of their radii, the circles are:
Answer: B — When d = |r1 − r2|, the smaller circle sits inside the larger one and just touches it at one point (internally tangent).
Q40. For two circles where one is inside the other without touching, how many common tangents exist?
Answer: A — When one circle is entirely inside another without touching, no common tangent can be drawn.
Q41. Which best describes the relationship between the tangent and the circle at the point of tangency?
Answer: C — "Tangere" in Latin means "to touch." The tangent line grazes the circle at one point and does not cut or enter it.
Q42. In the cam drawing, centres C3 and C4 are found by intersecting arcs from C1 and C2. This is the same principle as:
Answer: B — The cam's curved outline uses the same "sum of radii" arc-intersection method, ensuring smooth tangential transitions — the same as Exercise 1.9.
Q43. How many common tangents exist for two circles that touch each other externally?
Answer: B — When two circles touch externally: 2 external tangents + 1 at the touching point = 3 total.
Q44. Which instrument is primarily used to draw tangent lines perpendicular to a radius in a basic tangent construction?
Answer: C — A set square is placed against the radius line OP to draw the perpendicular tangent at P.
Q45. The tangent–radius perpendicularity rule means that, to verify a tangent construction, you can:
Answer: B — The easiest verification is to check the 90° angle between radius OP and the tangent line at P using a protractor or set square.
Q46. In engineering drawing, a "common external tangent" to two pulleys defines:
Answer: B — An open belt drive uses external common tangents. A cross belt drive uses internal common tangents, reversing rotation direction.
Q47. When drawing a tangent from an external point to a circle, how many tangents can be drawn from that single point?
Answer: B — From any external point, exactly 2 tangent lines can be drawn to a circle (one on each side). Both are equal in length from the external point.
Q48. Which statement about the normal to a circle is TRUE?
Answer: C — The normal at any point P on a circle is perpendicular to the tangent at P and passes through the centre O (it is the radius OP extended).
Q49. In Ex.1.9, C3 and C4 are found above and below the centreline. The final tangential arcs are drawn from C3/C4. The distance from C3 to C1 equals:
Answer: C — C3 was located by intersecting arcs of 35 mm from C1 and 30 mm from C2. Distance C3–C1 = 35 mm = R1 + R_arc = 15 + 20, guaranteeing external tangency.
Q50. Two circles intersecting at two points can have how many common tangents?
Answer: C — When two circles overlap (intersect at 2 points), only 2 external common tangents can be drawn. No internal tangents are possible in this configuration.
5
Speed & Pattern Recognition — Hover to Reveal
Rapid Fire MCQs
50 quick-fire questions. Hover over the answer column to reveal each answer. Aim for <30 seconds per question.
| # | Question | A | B | C | D | Ans |
|---|---|---|---|---|---|---|
| 1 | A tangent touches a circle at how many points? | Zero | One | Two | Three | B |
| 2 | Angle between radius and tangent at point of tangency: | 45° | 60° | 90° | 180° | C |
| 3 | Touch point of a tangent on a circle is called: | Origin | Focus | Vertex | Point of tangency | D |
| 4 | A secant differs from a tangent by: | Being shorter | Touching at 2 points | Passing through centre | Being curved | B |
| 5 | Circles sharing the same centre are called: | Eccentric | Tangential | Concentric | Common | C |
| 6 | To draw a tangent at P, radius OP is drawn _____ to the tangent. | Parallel | Perpendicular | Equal | Tangent | B |
| 7 | External tangents lie _____ both circles. | Inside | Between | Outside | Through | C |
| 8 | Internal tangents pass _____ the two circles. | Through | Between | Around | Outside | B |
| 9 | Auxiliary circle for external tangent has dia = D1 _____ D2. | Plus | Times | Minus | Divided by | C |
| 10 | In Ex.1.7 (dia 40 and 30), auxiliary circle diameter is: | 70 mm | 10 mm | 20 mm | 35 mm | B |
| 11 | In Ex.1.8, point A is the _____ of C1C2. | Bisector | Centre | Midpoint | End point | C |
| 12 | In Ex.1.8, B is found by bisecting: | C1C2 | C1A | AB | AC2 | B |
| 13 | Internal tangents for equal circles in Ex.1.8 are named: | S1 and S2 | T1 and T2 | P1 and P2 | N1 and N2 | B |
| 14 | In Ex.1.9, circles have diameters 30 and 20. Radii are: | 30 and 20 | 10 and 15 | 15 and 10 | 20 and 30 | C |
| 15 | Arc from C1 in tangential curve construction (R=20 arc) has radius: | 15 | 20 | 30 | 35 | D |
| 16 | Arc from C2 in tangential curve construction has radius: | 10 | 20 | 30 | 35 | C |
| 17 | The final arcs in Ex.1.9 are drawn from C3/C4 with radius: | 15 | 10 | 20 | 25 | C |
| 18 | In Ex.1.6, radii for 3 mutually tangential circles are found using: | Medians | Altitudes | Angle bisectors | Perpendicular bisectors | C |
| 19 | Radius of circle at centre A in Ex.1.6 is: | AB | AF | AC | AG | B |
| 20 | A line perpendicular to the tangent at its point of tangency is called the: | Secant | Chord | Normal | Bisector | C |
| 21 | For 2 separate circles, total common tangents (ext + int) = | 2 | 3 | 4 | 6 | C |
| 22 | Ball bearings are a practical example of _____ circles. | Eccentric | Tangential | Concentric | External | C |
| 23 | In Ex.1.1, the tangent at P is named: | OP | RS | AB | PQ | B |
| 24 | Two circles externally touching each other have ___ common tangents. | 4 | 2 | 3 | 1 | C |
| 25 | One circle inside another without touching has ___ common tangents. | 0 | 1 | 2 | 4 | A |
| 26 | Open belt drive layout uses _____ tangents. | Internal | External | Radial | Normal | B |
| 27 | Cross belt drive layout uses _____ tangents. | External | Normal | Internal | Eccentric | C |
| 28 | In Ex.1.3, the centre O is at intersection of two _____ to the given lines. | Perpendiculars | Bisectors | Parallel lines | Normals | C |
| 29 | In Ex.1.2, point O is found by drawing arcs from: | A and B | E and F | M and N | C1 and C2 | B |
| 30 | From an external point to a circle, _____ tangents can be drawn. | One | Two | Three | Four | B |
| 31 | Centre distance in Ex.1.8 is: | 40 mm | 30 mm | 60 mm | 80 mm | C |
| 32 | Diameter of each circle in Ex.1.8 is: | 30 mm | 60 mm | 40 mm | 20 mm | C |
| 33 | In Ex.1.7, the external tangent points are named: | A and B | E and F | P1 and P2 | C3 and C4 | C |
| 34 | For external tangent construction, C2P2 is drawn _____ to C1P1. | Perpendicular | Equal | Parallel | At 45° | C |
| 35 | The connecting arc in Ex.1.9 is _____ to both circles. | Secant | Normal | Tangential | Eccentric | C |
| 36 | A half-circle is called a: | Quarter arc | Semi-circle | Tangential arc | Chord arc | B |
| 37 | Formula for external tangency locus: arc radius from C1 = R_arc _____ R1 | Minus | Divided by | Plus | Times | C |
| 38 | Internal tangents cross the line C1C2 at its: | One-third point | End point | Midpoint | Centre of curvature | C |
| 39 | Two circles that intersect at two points can have how many common tangents? | 4 | 3 | 2 | 0 | C |
| 40 | Tangential arc from C3 touching circle D1 (R=15): distance C3–C1 must equal: | 15 | 20 | 30 | 35 | D |
| 41 | Circles within a circle with different centres are called: | Concentric | Eccentric | Tangential | Parallel | B |
| 42 | In Ex.1.6, triangle ABC is formed by: | Bisectors of angles | Lines joining three circle centres | Tangent lines | Medians | B |
| 43 | Radius of circle at centre C in Ex.1.6 is: | CF | CB | CA | CG | D |
| 44 | The "normal" and "tangent" at a point on a circle are always: | Parallel | Equal in length | Perpendicular to each other | Concurrent | C |
| 45 | In Ex.1.9, centres C3 and C4 are located _____ the horizontal centreline. | Left and right | Above and below | On the circle | At the tangent points | B |
| 46 | Outer circle of a ball bearing is _____ to the inner one. | Tangential | Eccentric | Concentric | Secant | C |
| 47 | In all tangent constructions, the key first step is always to establish: | The chord through the point | The radius to the tangent point | An auxiliary concentric circle | The midpoint of the circle | B |
| 48 | For Ex.1.7, the tangent from C2 to the auxiliary circle means the angle at P is: | 45° | 60° | 90° | 30° | C |
| 49 | Best verification after tangent construction: check angle between _____ and tangent at P is 90°. | Chord OP | Diameter | Radius OP | Normal to chord | C |
| 50 | Two identical circles touching externally have their common internal tangent at: | Combined centre | Point of tangency | Radius midpoint | Chord intersection | B |
📋 Rapid Fire — Answer Key
1. B
2. C
3. D
4. B
5. C
6. B
7. C
8. B
9. C
10. B
11. C
12. B
13. B
14. C
15. D
16. C
17. C
18. C
19. B
20. C
21. C
22. C
23. B
24. C
25. A
26. B
27. C
28. C
29. B
30. B
31. C
32. C
33. C
34. C
35. C
36. B
37. C
38. C
39. C
40. D
41. B
42. B
43. D
44. C
45. B
46. C
47. B
48. C
49. C
50. B
6
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