In an isosceles triangle at an apex of 90 degrees, the lateral side is 4. Find the length of the median drawn to this side.

Since the triangle is isosceles, and the angle B = 90, then the angle BAC = BCA = (180 – 90) / 2 = 45.

CM is the median of the triangle, then AM = BM = AB / 2 = 4/2 = 2 cm.

Let us define the hypotenuse AC by the Pythagorean theorem.

AC ^ 2 = AB ^ 2 + BC ^ 2 = 2 * AB ^ 2 = 2 * 16 = 32.

AC = 4 * √2 cm.

From the triangle AMC, by the cosine theorem, we determine the length of the median CM.

CM ^ 2 = AM ^ 2 + AC ^ 2 – 2 * AM * AC * Cos45 = 4 + 32 – 2 * 2 * 4 * √2 * √2 / 2 = 36 – 16 = 20.

CM = √20 = 2 * √5 cm.

Answer: The length of the median is 2 * √5 cm.