From the apex of an isosceles triangle, a median of 4 cm is drawn. This median cuts off a triangle

From the apex of an isosceles triangle, a median of 4 cm is drawn. This median cuts off a triangle with a perimeter of 12 cm and an area of 6 cm2. Calculate the length of the base and side of an isosceles triangle.

Consider an isosceles triangle ABC with lateral sides AB = BC and base AC.

Let BM be the median of triangle ABC, i.e. AM = CM.

By the condition of the problem, it is known that BM = 4 cm,

the perimeter P of triangle ABM is 12 cm,

the area S of the triangle ABM is 6 cm2.

Since the triangle is isosceles, the median BM is also the height. Therefore, triangle ABM is rectangular and its area is:

S = 1/2 * AM * BM,

6 = 1/2 * AM * 4,

AM = 3 cm.

Hence, AC = AM + CM = 2 * AM = 2 * 3 = 6.

By the Pythagorean theorem we have:

AB ^ 2 = AM ^ 2 + BM ^ 2 = 3 ^ 2 + 4 ^ 2 = 25.

AB = 5.

Note that we obtained the lengths AC = 6 cm, AB = BC = 5 cm, without using the fact that the perimeter of ABM is 12.