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प्रश्न
In ∆MNP, NQ is a bisector of ∠N. If MN = 5, PN = 7 MQ = 2.5 then find QP.
उत्तर
In ∆MNP, NQ is the bisector of ∠N. ...(given)
∴ `("PN")/("MN") = ("QP")/2.5` ...(Property of angle bisector of a triangle)
∴ `7/5 = ("QP")/2.5`
∴ QP = `((7 xx 2.5))/5`
∴ QP = 3.5 units
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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]
