Let $\triangle abc$ be a right triangle such that $b$ is a right angle. a circle with diameter of $bc$ meets side $ac$ at $d.$ if $ad = 1$ and $bd = 4,$ then what is $cd$?

16

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Explanation:

Draw the diagram or use the diagram provided with your book. The diagram should look something like what you see in the attached image (see below). 

Given Lengths:
AD = 1
BD = 4
CD = x
we want to solve for x to find the length of CD

Given Angles
Angle ABC = 90 degrees

Note how point D is on the circle. Angle BDC must be 90 degrees through the converse of Thales' Theorem. This is a special case of the Inscribed Angle Theorem.

Because angle BDC is a right angle, we have three similar triangles. The three triangles are:
Triangle ABC = largest triangle
Triangle ABD = smaller triangle on the left (inside largest)
Triangle BDC = smaller triangle on the right (inside largest)

The similar triangles allow us to state that the corresponding sides are proportional, therefore leading to this equation

AD/BD = BD/CD

Plug in the given values and solve for x
AD/BD = BD/CD
1/4 = 4/x
1*x = 4*4 <--- cross multiply
x = 16

So CD = 16