9514 1404 393
Answer:
(d) AE = (32√6)/3
Step-by-step explanation:
The side length ratios in a 30-60-90 triangle are 1 : √3 : 2. In a 45-45-90 triangle, they are 1 : 1 : √2.
So, we have the side length ratios ...
CE : CD = 2 : √3
BE : CE = √2 : 1
AE : BE = 2 : √3
Then the ratio AE : CE is ...
[tex]\dfrac{AE}{CD}=\dfrac{2}{\sqrt{3}}\cdot\dfrac{\sqrt{2}}{1}\cdot\dfrac{2}{\sqrt{3}}=\dfrac{4\sqrt{2}}{3}[/tex]
Multiplying by the length of CD, we have ...
AE = (8√3)(4√2)/3
AE = (32√6)/3 . . . . . matches the last choice
