In rectangle ABCD, sides AB = 35, AD = 12. The bisector of angle ABD intersects line CD at point E

| April 11, 2021 | education

In rectangle ABCD, sides AB = 35, AD = 12. The bisector of angle ABD intersects line CD at point E, and the bisector of angle ADB intersects line BC at point F. Find the squared length of segment EF.

Let us determine, by the Pythagorean theorem, the length of the diagonal BD of the rectangle.

BD ^ 2 = AB ^ 2 + AD ^ 2 = 35 ^ 2 + 12 ^ 2 = 1225 + 144 = 1369.

ВD = 37 cm.

Angle BDF = EBD, since EB is a bisector. Angle ADF = BDF, as the angles lying crosswise at the intersection of straight lines AD and CF of secant DF, then angle BDF = DBF, and therefore triangle BDF is isosceles, BD = BF = 37 cm.

Then the length of the segment CF = CB + BF = 12 + 37 = 49 cm.

Angle ABD = ADF, since DF is a bisector. Angle ABE = BED, as the angles lying crosswise at the intersection of straight lines EC and AB secant EB, then the angle BDE = DEB, and therefore the triangle EDB is isosceles, ED = BD = 37 cm.

Then the length of the segment CE = CD + ED = 35 + 37 = 72 cm.

From the right-angled triangle ECF, by the Pythagorean theorem, EF ^ 2 = CE ^ 2 + CF ^ 2 = 72 ^ 2 + 49 ^ 2 = 5184 + 2401 = 7585.

Answer: The square of the length of the segment EF is 7585 cm.



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