The lengths of the medians through acute angles of a right angled triangle are 3 and 4. Its area is -

The area of a right-angled triangle can be calculated using the formula: 1/2 * base * height.

In this case, the medians through the acute angles of the triangle are given as 3 and 4. In a right-angled triangle, the median through the right angle is half the hypotenuse. Therefore, the hypotenuse is twice the median through the right angle.

However, in this problem, we are given the lengths of the medians through the acute angles, not the right angle. The lengths of the medians through the acute angles of a right-angled triangle are related to the lengths of the sides of the triangle.

The median to the hypotenuse of a right triangle is half the length of the hypotenuse. Therefore, if we let the hypotenuse be 2a, then the medians are a√5 and a√5/2.

Setting these equal to the given lengths, we have:

a√5 = 4 and a√5/2 = 3

Solving these equations for a gives a = 4/√5 = 8/5 and a = 6/√5 = 12/5.

These are not equal, so there seems to be a mistake in the problem as stated. The lengths of the medians through the acute angles of a right-angled triangle cannot be 3 and 4.

If the lengths of the medians were given correctly, we could use them to find the lengths of the sides of the triangle, and then use those to calculate the area.