The acute angle of a right-angled triangle is 45 degrees. A middle line is drawn through the middle of the leg

The acute angle of a right-angled triangle is 45 degrees. A middle line is drawn through the middle of the leg, cutting off the trapezoid from the triangle, the area of which is 6. Find the height of this trapezoid.

Given:
right-angled triangle ABC,
angle С = 45 degrees,
HK – middle line,
AHKС – trapezoid,
S AHKС = 6.
Find HА -?
Decision:
1) Consider the triangle ABC. If the angle C = 45 degrees, then the angle B = 45 degrees, then the triangle ABC is also isosceles. Then AB = AC.
2) If НK is the middle line, then AH = 1/2 * AC = HK. Then AC = 2 * AH.
S AHKС = (HK + AC) / 2 * AH;
6 = (AH + 2 * AH) / 2 * AH;
6 = 3 * AН / 2 * AН;
6 = 3 AH ^ 2/2;
3 AH ^ 2 = 12;
AH ^ 2 = 12: 3;
AH ^ 2 = 4;
AH = 2.
Answer: 2.