Triangle ABC, angle C = 90 degrees, angle B = 45 degrees, AB = 8cm, BM-median, find BM.

This is not directly stated in the problem, but triangle ABC, besides being rectangular, is also isosceles. Let’s prove.
Since one angle is 90 °, the second is 45 °, then we find the third angle A:
180 ° – 90 ° – 45 ° = 45 °.
And if the two angles are the same, then the triangle is isosceles. It follows from this that CA = CB.
Let’s find each of these sides. Since they are equal, and we know the hypotenuse, using the Pythagorean theorem we can find AC and СВ:
AC² + СV² = AB²;
AC² + AC² = 8²;
2AC² = 64;
AC² = 64: 2;
AC² = 32.
AC = √32;
AC = 4√2.
BM² = (2√2) ² + (4√2) ²;
BM² = 40;
BM = √40;
ВM = 2√10.
ANSWER: ВM = 2√10 cm.