In rectangle ABCD – the bisector of angle A intersects BC at point K. Find AK if AD = 11, perimeter ABCD = 38.

1. In a rectangle, the sides opposite each other are equal. Using the formula for calculating the perimeter of a rectangle, we calculate the length of the side AB:

2AB + 2AD = 38;

2AB = 38 – 2AD = 38 – 22 = 16 cm.

AB = 8 cm.

2. Angle BАК = 90 °: 2 = 45 °, since the bisector divides the angle in half.

4. Angle AKB = 180 ° – 90 ° – 45 ° = 45 °.

5. Triangle AKB is isosceles, since the angles at the base of AB are equal. AB = BK = 8 cm.

6. AK ^ 2 = AB ^ 2 + BK ^ 2;

AK = √AB ^ 2 + BK ^ 2 = √8 ^ 2 + 8 ^ 2 = 8√2 cm.

Answer: AK length is 8√2 cm.