Here we will prove that two circles with centres X and Y touch externally at T. A straight line is drawn through T to cut the circles at M and N. Proved that XM is parallel to YN.
Solution:
Given: Two circles with centres X and Y touch externally at T. A straight line is drawn through T to cut the circles at M and N.
To prove: XM ∥ YN.
Construction: Join T to X and Y.
Proof:
Statement |
Reason |
1. In ∆XMT, ∠XMT = ∠XTM |
1. XM = XT, being radii. |
2. In ∆YNT, ∠YNT = ∠YTN |
2. YN = YT, being radii. |
3. XTY is a straight line. |
3. The point of contact of two circles lies on the straight line joining their centres. |
4. ∠XTM = ∠YTN |
4. Vertically opposite angles. |
5. ∠XMT = ∠YNT |
5. From statements 1, 2 and 4. |
6. XM ∥ YN. (Proved) |
6. Alternate angles are equal, using statement 5. |
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