Today our topic of discussion is Internal Division of a Line Segment in definite ratio.
Internal Division of a Line Segment in definite ratio
Internal Division of a Line Segment in definite ratio
If A and B are two different points in a plane and m and n are two natural numbers, we acknowledge that there exists a unique point X lying between A and B and AX / X * B = m : n
In the above figure, the line segment AB is divided at X internally in the ratio m: nie. AX / X * B = m / n
Construction 12. To divide a given line segment internally in a given ratio.
Special Nomination: Let the line segment AB be divided internally in the ratio m: n.
Drawing: Let an angle BAX be drawn at A. From AX cut the lengths AE = m and EC = n sequentially. Join B, C. At E, draw line segment ED parallel to CB which intersects AB at D. Then the line segment AB is divided at D internally in the ratio m: n.
Proof: Since the line segment DE is parallel to a side BC of the triangle ABC. . AD / D * B = AE / E * C = m / n
Work: Divide a given line segment in definite ratio internally by an alternative method.
Example 1. Divide a line segment of length 7 cm internally in the ratio 3 : 2.
Solution: Draw any ray AG. From AG, cut a line segment AB = 7cm Draw an angle ∠BAX at A. From AX, cut the lengths AE = 3 c.m. and EC = 2cm from EX. Join B, C At E, draw an ∠AED equal to∠ACB whose side intersects AB at D. Then the line segment AB is divided at D internally in the ratio 3: 2.
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