Here we will prove that if two circles touch each other, the point of contact lies on the straight line joining their centres.
Case 1: When the two circles touch each other externally.
Given: Two circles with centres O and P touch each other externally at T.
To prove: T lies on the line OP.
Construction: Draw a common tangent XY through the point of contact T. Join T to O and P.
Proof:
Statement |
Reason |
1. ∠OTX = 90° |
1. Radius OT ⊥ tangent XY. |
2. ∠PTX = 90° |
2. Radius PT ⊥ tangent XY. |
3. ∠OTX + ∠PTX = 180° ⟹ ∠OTP = 180° ⟹ OTP is a straight line ⟹ T lies on OP. (Proved) |
3. Adding statement 1 and 2. |
Case 2: When the two circles touch each other internally at T.
To prove: T lies on OP produced.
Construction: Draw a common tangent XY through the point of contact T. Join T to O and P.
Proof:
Statement |
Reason |
1. ∠OTX = 90° |
1. Radius OT ⊥ tangent XY. |
2. ∠PTX = 90° |
2. Radius PT ⊥ tangent XY. |
3. OT and PT are both ⊥ to XY at the same point T. |
3. From statement 1 and 2. |
4. OT and PT lies on the same straight line ⟹ OTP is a straight line ⟹ T lies on OP. (Proved) |
4. Only one perpendicular can be drawn to a line through a point on it. |
Note: Let two circles with centres O and P touch each other at T. Let OT = r1 and PT = r2 and r1 > r2.
Let the distance between their centres = OP = d.
It is clear from the figures that
• When the circles touch externally, d = r_{1} + r_{2}.
• When the circles touch internally, d = r_{1} - r_{2}.
From Two Circles Touch each Other to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.