On the side BC of rectangle ABCD, in which AB = 60 and AD = 71, point E is marked so that ∠EAB = 45. Find ED.

A rectangle is a rectangle with equal and parallel opposite sides:

AB = CD = 60;

AD = DC = 71;

Angle A = 90 °,

Since we know the angle EAD = 45 °, we find the angle EAB:

∠ EAB = ∠ A – ∠ EAD = 90 ° – 45 ° = 45 °.

The sum of the angles in any triangle is 180 °. We know two angles in a right-angled triangle ABE, so we can find the angle AEB:

∠ AEB = 180 ° – ∠ B – ∠ EAB = 180 ° – 90 ° – 45 ° = 45 °.

The ABE triangle is isosceles, since ∠ EAB = ∠ AEB = 45 °. So its sides are also equal:

AB = BE = 60;

Then EC = BC – BE = 71 – 60 = 11.

Find the hypotenuse of the right-angled triangle ECD by the Pythagorean theorem:

EC² + CD² = ED²;

11² + 60² = ED²;

121 + 3600 = ED²;

ED² = 3721;

ED = √3721;

ED = 61

Answer: 61