45-45-90 triangle with 30-60-90 triangle

[coooool222]

I also have another question about this triangle. The Hypotenuse of the the 45-45-90 triangle is 5 sqrt of 2 since i multiplied it with sqrt of 2. Does this mean that the shorter leg of the 30-60-90 triangle is 5 sqrt of 2 over sqrt of 3. Does it also mean that x (which is the hypotenuse of the 30-60-90 triangle) is 10 sqrt of 2 over sqrt of 3 because I multiplied it with 2?

 

[Steven G]

Please show us your work as you do have some errors.

 

[pka]

To co12: Not Really! Any \(30-60-90\) right triangle is "one half" an equilateral triangle.
Thus the length of the hypotenuse the \(45-45-90\) right triangle is \(5\sqrt 2\).
So the third side of the \(30-60-90\) right triangle has length \(\dfrac{x}{2}\).

 

[coooool222]

To co12: Not Really! Any \(30-60-90\) right triangle is "one half" an equilateral triangle.
Thus the length of the hypotenuse the \(45-45-90\) right triangle is \(5\sqrt 2\).
So the third side of the \(30-60-90\) right triangle has length \(\dfrac{x}{2}\).

i am really confused i thought x is 10 sqrt of 2 over sqrt of 3 since the shortest side of the 30-60-90 triangle is 5 sqrt of 2 over sqrt of 3?

 

[pka]

i am really confused i thought x is 10 sqrt of 2 over sqrt of 3 since the shortest side of the 30-60-90 triangle is 5 sqrt of 2 over sqrt of 3?

In any \(30,~60,~90\) triangle the three sides have lengths \(2y,~y,~\&~y\sqrt 3\).
In any \(45,~45,~90\) triangle the three sides have lengths \(z,~z,~\&~z\sqrt 2\).

 

[Dr.Peterson]

i am really confused i thought x is 10 sqrt of 2 over sqrt of 3 since the shortest side of the 30-60-90 triangle is 5 sqrt of 2 over sqrt of 3?

It's hard to read what you are writing, which may be part of the confusion. Otherwise, I can't explain why they think you are wrong, since what they have said agrees with you.

You are right that the shared edge is [MATH]5\sqrt{2}[/MATH], which you can write as 5 sqrt(2). Then the short side of the 30-60-90 triangle is that divided by sqrt(3), [MATH]\frac{5\sqrt{2}}{\sqrt{3}}[/MATH]; and then x is twice that, [MATH]\frac{10\sqrt{2}}{\sqrt{3}}[/MATH], which you could write as 10 sqrt(2)/sqrt(3). Of course, you may be expected to rationalize the denominator.