# Through the point M of the side КР of the triangle FKP, a straight line is drawn parallel to the side FK and intersecting

Through the point M of the side КР of the triangle FKP, a straight line is drawn parallel to the side FK and intersecting the side FP to the point T. Find TM if FK = 52 cm, FT = 12 cm, TP = 36 cm.

Let us prove that the triangle TPM is similar to the triangle FKP.

The angle P is common for the triangles.

The segment ТМ, by condition, is parallel to FK, then the angle КFP = МТР as the corresponding angles at the intersection of parallel straight lines FK and ТМ secant FP.

Then triangles TPM and FKP are similar in two angles.

Side length FP = FT + TP = 12 + 36 = 48 cm.

From the similarity of triangles: TM / FK = TP / FP.

TM = FK * TP / FP = 52 * 36/48 = 39 cm.

Answer: The length of the TM segment is 39 cm.

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