# In quadrilateral ABCD, BC is parallel to AD. E is the foot of the perpendicular from B to AD. Find BE, if AB = 17cm, BC = 16cm, CD = 25cm and AD = 44cm

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Since the side BC is parallel to AD, that means that the length of the perpendicular from B to AD is equal to the perpendicular from D to BC.

We'll create two right angle triangles ABE and DFC, where BE = DF.

Let the leg AE = x. We'll write Pythagorean theorem in triangle ABE, where AB = 17 is hypotenuse.

BE^2 = 17^2 - x^2

Let the leg FC = 44-16-x = 28-x

We'll write Pythagorean theorem in triangle DFC, where DC = 25 is hypotenuse.

CF^2 = 25^2 - (28-x)^2

But CF = BE=>17^2 - x^2 = 25^2 - (28-x)^2

We'll move the terms in x to the left side:

(28-x)^2 - x^2 = 25^2 - 17^2

We'll expand the binomial:

28^2 - 56x + x^2 - x^2 = 25^2 - 17^2

The difference of two squares form the right side, returns the product:

28^2 - 56x = (25-17)(25+17)

28^2 - 56x = (8)(42)

56x = 784 - 336

56x = 448

x = 8

Now, we'll calculate BE^2 = (17+x)(17-x)

BE^2 = 25*9 = 225

BE = 15

**The requested length of the perpendicular drawn from B to AD, isĀ BE = 15.**