In a right-angled triangle ABC, angle C = 90 degrees, angle A = 30 degrees, CВ = 6cm.

In a right-angled triangle ABC, angle C = 90 degrees, angle A = 30 degrees, CВ = 6cm. Find the perimeter of a triangle whose vertices are the midpoints of the sides of this triangle.

Each of the sought sides of the triangle KMН is the middle line of the triangle ABC, since points M, H, K are the middle of the segments AB, BC and AC.

The CB leg lies against an angle of 30, then AB = 2 * BC = 2 * 6 = 12 cm.

AC ^ 2 = AB ^ 2 – BC ^ 2 = 144 – 36 = 108.

AC = 6 * √3 cm.

Then KM = BC / 2 = 6/2 = 3 cm.

KН = AB / 2 = 12/2 = 6 cm.

MH = AC / 2 = 6 * √3 / 2 = 3 * √3 cm.

Then Ркмн = 3 + 6 + 3 * √3 = 9 + 3 * √ 3 cm.

Answer: The perimeter of the triangle is 9 + 3 * √3 cm.