Point n lies on the side ab in △abc. points k and m are midpoints of segments
bc
and
an
respectively. segments
cn
and
mk
intersect at point t. it is known that ∠ctk≅∠tmn and ab=7. find the length of segment
cn
.

  CN = 7

Step-by-step explanation:

In the attached figure, we have drawn line CD parallel to AB with D a point on line MK. We know ΔMNT ~ ΔDCT by AA similarity, and because of the given angle congruence, both are isosceles with CD = CT. Likewise, we know ΔCDK is congruent to ΔBMK by AAS congruence, since BK = CK (given).

Then CD = BM (CPCTC). Drawing line NE creates isosceles ΔNEC ~ ΔTDC and makes CE = AB. Because ΔNEC is isosceles, CN = CE = AB = 7.

The length of segment CN is 7.

_____

If you assume CN is constant, regardless of the location of point N (which it is), then you can locate point N at B. That also collocates points T and K and makes ΔBMK both isosceles and similar to ΔBAC. Then CN=AB=7.

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