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| Section Formula |
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| To find the co-ordinates of a point C (x, y) which divides the line segment joining the two points A (x1,y1) and B (x2,y2) in the ratio m:n ( internally and externally). |
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| Case (i) |
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| C divides AB internally. |
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Let A (x1, y1) and B (x2, y2) be the two points joined by line segment
AB. Let C (x, y) be the point on the line segment such that  |
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| (In this case, AC and CB are real in the same direction on the line AB.) |
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| Draw AP, CR and BQ perpendicular to x-axis. |
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| AM perpendicular to CR and CM perpendicular to BQ. |
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| AM = PR = x-x1 |
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| CN = RQ = x2-x |
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| CM = y-y1 |
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| BN = y2-y |
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| From the similar triangles, CAM and BCN, we have |
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| Case (ii) |
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| C divides AB externally. |
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| From the similar triangles, CAM and CBN, we have |
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| If i) r>0, C divides internally. |
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| ii) r<0, C divides externally. |
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| Note: |
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| If C divides internally in the ratio 1:1 i.e., C is the midpoint of AB, then |
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| This formula is called the midpoint formula. |
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| Example: |
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| Find the coordinates of the points A (-3, -4), B (-8,7) which divides the line segment joining the points A and B in the given ratio 5:7 |
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| i) internally and |
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| ii) externally. |
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| Suggested answer: |
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| x1 = -3, x2 = -8 |
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| y1 = -4, y2 = 7 |
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| m = 5, n = 7 |
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| i) Internal division: |
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| ii) External division: |
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