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Touching Circles
External contact: distance between centres = r1 + r2. Internal contact: distance between centres = |r1 - r2|. Two congruent circles touching externally have d = 2r. Distinct congruent circles cannot touch internally; if their centres coincide, they are the same circle and have infinitely many common points.
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Touching Circles
What is it?
External contact: distance between centres = r1 + r2. Internal contact: distance between centres = |r1 - r2|. Two congruent circles touching externally have d = 2r. Distinct congruent circles cannot touch internally; if their centres coincide, they are the same circle and have infinitely many common points.
In simple words
- External contact: distance between centres = r1 + r2.
- EXTERNAL: Radii 2 and 1: d=3.
- INTERNAL: Larger minus smaller: 2.5-1.4=1.1.
- Worked-method reminder: d = 1.1 cm
Important terms
TOUCHING CIRCLES: External contact: distance between centres = r1 + r2. Internal contact: distance between centres = |r1 - r2|. Two congruent circles touching externally have d = 2r. Distinct congruent circles cannot touch internally; if their centres coincide, they are the same circle and have infinitely many common points.
EXTERNAL: Radii 2 and 1: d=3. Radii 1.4 and 2.5: d=3.9. The centres and the point of contact are collinear.
INTERNAL: Larger minus smaller: 2.5-1.4=1.1. One circle inside the other, touching at one point. Do not add for internal contact.
What you should learn
- Explain touching circles using the correct definitions, conditions, and notation.
- Apply external: sum of radii, internal: difference correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- 2+1=3
- 1.4+2.5=3.9
- Equal r: d=2r
EXTERNAL
Radii 2 and 1: d=3. Radii 1.4 and 2.5: d=3.9. The centres and the point of contact are collinear.
INTERNAL
Larger minus smaller: 2.5-1.4=1.1. One circle inside the other, touching at one point. Do not add for internal contact.
Recommended for Class 10
Full school-level explanation
External contact: distance between centres = r1 + r2. Internal contact: distance between centres = |r1 - r2|. Two congruent circles touching externally have d = 2r. Distinct congruent circles cannot touch internally; if their centres coincide, they are the same circle and have infinitely many common points.
Add or subtract the radii:
WHAT — TOUCHING CIRCLES
External contact: distance between centres = r1 + r2. Internal contact: distance between centres = |r1 - r2|. Two congruent circles touching externally have d = 2r. Distinct congruent circles cannot touch internally; if their centres coincide, they are the same circle and have infinitely many common points. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Touching Circles matters because tangent and circle-angle theorems explain the geometry at a point of contact. They support proofs, numerical angle questions, and practical circle constructions.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through EXTERNAL, then INTERNAL. Worked: radii 2.5 cm and 1.4 cm, internal contact. Find d. internal contact uses the difference d = 2.5 - 1.4 d = 1.1 cm Finish by checking signs, restrictions, units, and whether the answer matches the question.
EXTERNAL
Radii 2 and 1: d=3. Radii 1.4 and 2.5: d=3.9. The centres and the point of contact are collinear.
Remember
- d = r1+r2
Examples
- 2+1=3
- 1.4+2.5=3.9
- Equal r: d=2r
INTERNAL
Larger minus smaller: 2.5-1.4=1.1. One circle inside the other, touching at one point. Do not add for internal contact.
Remember
- d = |r1-r2|
Examples
- 2.5-1.4=1.1
- Not 3.9 for internal
- Congruent internal: d=0
Remember
- external: sum of radii
- internal: difference
- 1.1 cm
- sum vs difference
- d = r1+r2
- d = |r1-r2|
- external: sum of radii
- internal: difference
How the ideas connect
Worked: radii 2.5 cm and 1.4 cm, internal contact. Find d. internal contact uses the difference d = 2.5 - 1.4 d = 1.1 cm
Deeper understanding
Go beyond memorising Touching Circles. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: radii 2.5 cm and 1.4 cm, internal contact. Find d. internal contact uses the difference d = 2.5 - 1.4 d = 1.1 cm
Common mistakes
Exam focus
Core Idea
EXTERNAL
INTERNAL
Worked Method
Exam focus
Quick revision
- external: sum of radii
- internal: difference
- 1.1 cm
- sum vs difference
- TOUCHING CIRCLES: External contact: distance between centres = r1 + r2. Internal contact: distance between centres = |
- EXTERNAL: Radii 2 and 1: d=3. Radii 1.4 and 2.5: d=3.9. The centres and the point of contact are collinear.
Do not call coincident congruent circles 'internally tangent'. Tangent circles have exactly one common point; coincident circles have infinitely many.
For long answers on Touching Circles: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- external: sum of radii
- internal: difference
- 1.1 cm
- sum vs difference
Important formulas
- external: sum of radii
- internal: difference
- 1.1 cm
Common exam traps
- Do not call coincident congruent circles 'internally tangent'. Tangent circles have exactly one common point; coincident circles have infinitely many.
- Not 3.9 for internal
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Do not call coincident congruent circles 'internally tangent'. Tangent circles have exactly one common point; coincident circles have infinitely many.
- Mistake
Not 3.9 for internal
- Mistake
Do not treat EXTERNAL and INTERNAL as the same step; each has a different purpose.
- Mistake
Do not call coincident congruent circles 'internally tangent'. Tangent circles have exactly one common point; coincident circles have infinitely many.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
external: sum of radii
internal: difference
1.1 cm
sum vs difference
External contact: distance between centres = r1 + r2. Internal contact: distance between centres = |r1 - r2|. Two congruent circles touching externally have d =
d = r1+r2
- Definition
- External contact: distance between centres = r1 + r2. Internal contact: distance between centres = |r1 - r2|. Two congruent circles touching externally have d =
- Formula
- d = r1+r2
- Example
- 2+1=3
- Critical warning
- Do not call coincident congruent circles 'internally tangent'. Tangent circles have exactly one common point; coincident
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Exam Practice
Define
Q1. Define Touching Circles.
Explain
Q2. Explain: EXTERNAL.
Example
Q3. Give one clear example of Touching Circles and say why it fits.
Mistake
Q4. State one common mistake students make about Touching Circles.
Exam note
Q5. Write a short exam note on Touching Circles: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.