The medians of the triangle MNK meet at point O. Through point O. a straight line is drawn parallel to side MK

| February 18, 2021 | education

The medians of the triangle MNK meet at point O. Through point O. a straight line is drawn parallel to side MK and intersecting sides MN and MK at points A and B, respectively. Find MK if the length of line segment AB is 12 cm.

Let us prove that the triangles MND and ANO are similar.

The angle H is common for the triangles. Since, according to the condition, the segment AB is parallel to the segment MK, the angle MDN = AON as the corresponding angles at the intersection of parallel lines MK and AB secant ND. Then the triangles MND and ANO are similar in two angles.

By the property of the medians of the triangle, they are divided at the point O in the ratio 2/1.

Then HO = 2 * ND / 3.

AH / ND = 2/3 = K.

Then AO / MD = 2/3.

MD = 3 * AO / 2.

AO = AB / 2 = 12/2 = 6 cm.

MD = 3 * 6/3 = 6 cm.

ND is the median of the МНК triangle, then MK = 2 * MD = 2 * 9 = 18 cm.

Answer: The length of the MK segment is 18 cm.



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