On the BC side of the rectangle ABCD, in which AB = 24 and AD = 31, point E is marked

On the BC side of the rectangle ABCD, in which AB = 24 and AD = 31, point E is marked so that EAB = 45 degrees. Find ED

1. We calculate the value ∠АЕВ of the right-angled triangle ABE:

∠АЕВ = 180 ° – ∠АЕВ – ∠ВАЕ = 180 ° – 90 ° – 45 ° = 45 °.

2. The angles at the base AE of triangle ABE are equal (∠АЕВ = ∠ВАЕ = 45 °). Consequently,

the indicated triangle is isosceles. Hence, AB = BE = 24 units.

3. CE = BC – BE.

BC = AD = 31 units of measurement (opposite sides of the rectangle, according to its

properties are equal).

CE = 31 – 24 = 7 units.

4. Calculate the length of the segment DE:

DE = √CE² + CD² (along the tower of Pythagoras).

AB = CD = 24 units.

DE = √7² + 24² = √49 + 576 = √625 = 25 units.

Answer: the length of the segment DE is equal to 25 units of measurement.