Choose whether the given statement is true or false.Let X be any point on the side BC of a triangle ABC. If XM, XN are drawn parallel to BA and CA, BA in M, N respectively: MN meets BC produced in T then TX2=TB×TC. Is it true or False?

# Choose whether the given statement is true or false.Let X be any point on the side BC of a triangle ABC. If XM, XN are drawn parallel to BA and CA, BA in M, N respectively: MN meets BC produced in T then TX2=TB×TC. Is it true or False?

1. A
True
2. B
False

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### Solution:

It is being that XM is parallel to BA and XN is parallel to CA so, we can write it as: XM∥BA and  XN∥CA.

Consider  ΔTXM  here the line  XM∥BN  since line  XM∥BA, so by using the basic proportionality theorem we can write

Now we consider the  ΔTMC, here by using the basic proportionality theorem we can write

Now from the equations (i) and (ii), we can write

Hence by further solving the obtained equation we can say
TX2 = TB⋅TC
Hence proved that TX2=TB⋅TC. Hence Answer is True.

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