# In triangle ABC, segment MH is the middle line. AB = 20cm, BC = 24cm

In triangle ABC, segment MH is the middle line. AB = 20cm, BC = 24cm, AC = 18cm. MN is porallnaya AС. Find the perimeter of the MВН triangle.

First way.

Since, by condition, МН is the middle line of a triangle, its length is equal to half of the side of the triangle parallel to it. MH = AC / 2 = 18/2 = 9 cm.

Points M and H of the middle line of the triangle, respectively, divide the sides AB and BC in half, then:

AM = BM = AB / 2 = 20/2 = 10 cm.

CH = ВН = BC / 2 = 24/2 = 12 cm.

Then the perimeter of the triangle MВН = MН + MВ + ВН = 9 + 10 + 12 = 31 cm.

Triangles ABC and MBН are similar in two angles, angle C is common, angle BMН = BAC as the corresponding angles at the intersection of parallel straight lines MН and AC secant AB.

Let’s determine the coefficient of similarity of triangles.

Since MH is the middle line of the ABC triangle, MH = AC / 2.

K = AC / MH = 2.

The perimeter of the triangle ABC = AB = BC + CA = 20 + 24 + 18 = 62 cm.

The ratio of the perimeters of similar triangles is equal to the coefficient of similarity.

Then Рмвн = Равс / 2 = 62/2 = 31 cm.

Answer: The perimeter of the MВН triangle is 31 cm.

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