Through the middle O of the hypotenuse AB of the right-angled triangle ABC, straight lines are drawn parallel

| September 2, 2021 | education

Through the middle O of the hypotenuse AB of the right-angled triangle ABC, straight lines are drawn parallel to its legs. One of them crosses the leg AC at point M, the other – leg BC at point N. Find the hypotenuse AB if MN = 7 cm.

Let us prove that triangles CAB and CMN are similar, and hence
AB = 2 * MN.

Triangles CAB and MAO are similar, since angle C is equal to angle M, angle A is common, angle O is equal to angle B.
By the condition O, this is the middle of AB, which means AO = AB / 2.

Triangles MAO and CMN are equal, since NM is parallel to AC, MO is parallel to CB, and MN is parallel to AB, O is the middle of AB, and CMON is a rectangle, where MO = CN, CM = ON.
Thus, all angles and sides of triangles MAO and CMN are equal.

Therefore, MN = AO = AB / 2.
Hence, AB = 2 * MN = 2 * 7 = 14 cm.



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