Find the height of an isosceles right-angled triangle, dropped from the apex of the right angle, if the leg is 10cm.

(We denote the Vertex of the right angle by the letter A, the other two vertices B and C, the intersection point of the height and side BC by the letter D) By the Pythagorean theorem => BC ^ 2 = AB ^ 2 + AC ^ 2 => BC ^ 2 = 10 ^ 2 + 10 ^ 2 = 200 => BC = √ (200). Isosceles triangle => Height (Lowered from the top of the right angle) = median => BD = CD = √ (200) / 2 = √ (50) By the height theorem => AD ^ 2 = BD * CD => AD ^ 2 = √ (50) * √ (50) = √ (50) ^ 2 => AD = √ (50).
Answer: √ (50).