The two sides of the triangle are 30 and 40 cm. The length of the median drawn to the third side is 25 cm. Find the area of the triangle?

Let’s denote a triangle ABC, in it AB = 30 cm, BC = 40 cm, median BK = 25 cm.Let’s use the formula for the median in the triangle and find the third side of AC.
VK = 1 / 2√ (2AB² + 2BC² – AC²)
2BK = √ (2AB² + 2BC² – AC²)
4BK² = 2AB² + 2BC² – AC²
AC² = 2AB² + 2BC² – 4BK = 2 * 30² + 2 * 40² – 4 * 25² = 1800 + 3200 – 2500 = 2500;
AC = √2500 = 50 cm.
We know the sides, we find the semiperimeter of the triangle. Let us find the area by Heron’s formula.
p = 1/2 (30 + 40 + 50) = 60 cm.
S = √ p * (p – AB) * (p – BC) * (p – AC) = √ 60 * (60 – 30) * (60 – 40) * (60 – 50) = √ 60 * 30 * 20 * 10 = √ 360000 = 600 cm².
Answer: the area of the triangle is 600 cm².