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Question
From the information given in the figure, determine whether MP is the bisector of ∠KMN.
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Solution
We know that the bisector of an angle of a triangle divides the side opposite to the angle in the ratio of the remaining sides.
Now, `(KP)/(PN) = 2/3`
And `(MK)/(MN) = 5/6`
∵ `2/3 ≠ 5/6`
∴ `(KP)/(PN) ≠ (MK)/(MN)`
Hence, MP is not the bisector of ∠KMN.
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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]
