MathematicsChapter 10: Tangents and Angles of a Circle
Touching Circles
External contact: distance between centres = r1 + r2.
Learning objectives
- 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.
Definition
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.
Mathematical form
d = 2.5 - 1.4
d = 1.1 cm
centres = r1 + r2. Internal contact
centres =
Where / symbols
- 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.
Overview
Add or subtract the radii:
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.
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.
Examples 2.5-1.4=1.1 · Not 3.9 for internal · Congruent internal: d=0
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.
Worked example
- 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
In simple words
Do not call coincident congruent circles 'internally tangent'. Tangent circles have exactly one common point; coincident circles have infinitely many.
Key points
- external: sum of radii
- internal: difference
- d = r1+r2
- d = |r1-r2|
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Touching Circles
Sticky Notes · Quick Revision
Core Idea
- 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.
Worked Method
- 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 Misconception
- Do not call coincident congruent circles 'internally tangent'. Tangent circles have exactly one common point; coincident circles have infinitely many.
FBISE Exam Check
- Do not call coincident congruent circles 'internally tangent'. Tangent circles have exactly one common point; coincident circles have infinitely many.
60-second revision
- 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
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