In the rectangle ABCD, the bisector of angle A intersects the side BC at point K. Find the length

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

1. Using the formula for calculating the perimeter of a rectangle, we calculate the length AB:

2AB + 2AD = 38;

AB = (38 – 2AD) / 2 = (38 – 22) / 2 = 8 cm.

2. We calculate the value of the angle BАК, taking into account that the bisector divides the angle A in half:

90 °: 2 = 45 °.

2. The angle of the AKB is equal to:

180 ° – 90 ° – 45 ° = 45 °

3. Triangle AKB is isosceles, since the angles at the base of the AK are equal. Hence,

AB = BK = 8 cm.

4. Applying the Pythagorean theorem, we calculate the length of the AK:

AK = √AB ^ 2 + BK ^ 2 = √8 ^ 2 + 8 ^ 2 = √2 x 64 = 8√2 cm.

Answer: AK length is 8√2 cm.