ABCD is a convex cyclic quadrilateral such that AB=AD and ∠BAD=90∘. E is the foot of the perpendicular from A to BC, and F is the foot of the perpendicular from A to DC. If AE+AF=16, what is the area of ABCD?

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64

how 64 can you post process

To find the area of the convex cyclic quadrilateral ABCD, we will make use of the given information and some geometric properties.

Since ABCD is a cyclic quadrilateral, its opposite angles must add up to 180 degrees. Hence, we have:
∠ABC + ∠ADC = 180 degrees

Given that AB = AD, we can deduce that ∠ABC = ∠ADC.

Now, we are given that ∠BAD is a right angle, which implies that quadrilateral ABCD is an isosceles trapezoid. An isosceles trapezoid has two non-parallel sides of equal length.

Using this property, we conclude that BC = AD, and since AB = AD, we have BC = AB.

Now, let's denote BC = AB = x, AE = h, and DF = y.

Since ABCD is a cyclic quadrilateral, we can use the Pythagorean theorem in triangles ABE and ADF to establish the following relationships:

h^2 + x^2 = AB^2 and y^2 + x^2 = AD^2

From the given condition that AE + AF = 16, we can substitute the values of AE and AF to obtain:

h + y = 16

Now, let's solve this system of equations. Subtracting the second equation from the first, we get:

h^2 - y^2 = AB^2 - AD^2

(h + y)(h - y) = (AB + AD)(AB - AD)

Since AB = AD, we can simplify the equation to:

h + y = 2x

Substituting h + y = 16 (from the given information), we can solve for x:

16 = 2x

x = 8

Now that we have determined AB = BC = 8, we can calculate the area of the isosceles trapezoid ABCD using the formula:

Area = (1/2) * (AB + BC) * h

Area = (1/2) * (8 + 8) * h

Area = 8h

But we still need to find the value of h (which is AE). So let's use the given condition AE + AF = 16 to solve for h:

h + y = 16

Since BC = AB = 8, we can conclude that y = 8.

h + 8 = 16

h = 8

Finally, substituting the value of h into the area formula, we get:

Area = 8 * 8 = 64 square units

Therefore, the area of ABCD is 64 square units.