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Question
Given below is the triangle and length of line segments. Identify in the given figure, ray PM is the bisector of ∠QPR.
Solution
In ΔPQR,
`"PR"/"PQ" = 7/10` .....(i)
`"RM"/"QM" = 6/8` ......(ii)
From (i) and (ii),
`"PR"/"PQ" ≠ "RM"/"QM"`
∴ Ray PM is not the bisector of ∠QPR.
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solution:
In ∆PMQ,
Ray MX is the bisector of ∠PMQ.
∴ `("MP")/("MQ") = square/square` .............(I) [Theorem of angle bisector]
Similarly, in ∆PMR, Ray MY is the bisector of ∠PMR.
∴ `("MP")/("MR") = square/square` .............(II) [Theorem of angle bisector]
But `("MP")/("MQ") = ("MP")/("MR")` .............(III) [As M is the midpoint of QR.]
Hence MQ = MR
∴ `("PX")/square = square/("YR")` .............[From (I), (II) and (III)]
∴ XY || QR .............[Converse of basic proportionality theorem]
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