NotThatSmart

Let's graph the question. 

From the graph and the question, we know that

$CD = BD = 10 \\ ED = ED \\ \angle CDE = \angle BDE = 90$

So, by SAS congruence, we know that triangle CDE  is  congruent to triangle BDE

Thus, we can complete the problem in a series of steps using this congruence. 

First, we know that

$sin CED / CD = sin CDE / CE$

Plugging in some numbers, we find that 

$sin 75 / 10 = sin 90 / CE \\ CE = 10 sin 75 = BE ≈ 10.35$

Thus, 10.35 is approximately our answer. 

Thanks! :)

*1800 points

Let's note that $P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1$ is simply the perimeter of the decagon

We also know that $radius/2 ( -1 + \sqrt {5})$ is one of the sidelenghts. Thus, we can find the perimeter easily. 

The perimeter is just

$10 (1/2) ( -1 +\sqrt{ 5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18$

Thanks! :)

The length of $\overline{AB}$ is $4,$...Find the length $AB.$

The answer is 4. :)

$244224 \pmod {64} \equiv 0$

If angle XWZ = 60°, angle WXY = 180° - 60° = 120° because XY and WZ are parallels.

However, this means that Angle ZXW must be less than 120°.

In the problem though, it clearly defines ZXW as 120, meaning this problem is invalid. 

Thanks! :)