In an isosceles triangle ABC with the base AC, the medians intersect at point O. Find the area of triangle

In an isosceles triangle ABC with the base AC, the medians intersect at point O. Find the area of triangle ABC if OA = 13 cm, OB = 10 cm.

Let’s take advantage of the fact that the intersection point of the medians of the triangle divides each median into two parts in a ratio of 2: 1, counting from the vertex. It follows that:

| OD | = 0.5 | ОВ | = 0.5 * 10 = 5 and | BD | = | OB | + | OD | = 10 + 5 = 15.

Since triangle ABC is isosceles, the median BD is also the height,

those. BD perpendicular to AC.

Consider the triangle AOD. This is a right-angled triangle. Therefore, we can calculate

the length AD, knowing the lengths OA and OD by the Pythagorean theorem:

| AD | ^ 2 + | OD | ^ 2 = | AO | ^ 2 <=> | AD | ^ 2 = 13 ^ 2 – 5 ^ 2 = 12 ^ 2 <=> | AD | = 12

Then | AC | = 2 * | AD | = 24, because point D is the midpoint of side AC.

Knowing the length of the base and the height of the triangle, we find its area:

S = 0.5 * | AC | * | BD | = 0.5 * 24 * 15 = 180.

Answer: The area of ​​triangle ABC is 180.