Question

Find the value of x. If your answer is not an integer, leave it in simplest radical form. Diagram is not drawn to scale.. . (2 points).

Answer 1

You are supposed to already know the side lengths for these triangles in the Unit Circle. You need to take the time to memorize! 

Triangles are "similar" if their angle measures are the same but their side lengths are different (they will be proportional, though). These triangles are 45°-45°-90° and 30°-60°-90°, respectively. 

45°-45°-90°: 
........./| 
..1./....| [(√2)/2] 
/____| 
[(√2)/2] 

30°-60°-90°: 
..../| 
1/..| [(√3)/2] 
/__| 
..½ 

note that the hypotenuse-length is always 1 when the triangle is inscribed in the Unit Circle. Multiplying the lengths of *every* side of a triangle by the same number creates a Similar Triangle. 
========================= 
26) you know that one angle is 45°, and that this is a right triangle (because of the square in the corner). That means the other angle must be 45° and this is a well-known 45°-45°-90° triangle. You know the OPPOSITE side is 5 units long and the HYPOTENUSE is y units long. Therefore, for *this* triangle: 
sin(45°) = OPP/HYP = 5/y 

But from the Unit Circle, we also know the irreducible proportion for the SINE of a 45° angle is: 
sin(45°) = OPP/HYP = [(√2)/2]/(1) = (√2)/2. 

we can equate sin(45°) for similar triangles: 
sin(45°) = sin(45°) 
5/y = (√2)/2 
multiply by reciprocals or identity-fractions until you get 
y = 5√2 
===================== 
27) you know that one angle is 30°, and that this is a right triangle (because of the square in the corner). That means the other angle must be 60° and this is a well-known 30°-60°-90° triangle. You know the OPPOSITE side is x units long, the ADJACENT side is 3√3 units long, and the HYPOTENUSE is y units long. You can solve for x and y by relating the TANGENT and COSINE proportions for *this* triangle...: 
tan(30°) = OPP/ADJ = x/[(√3)/3] 
cos(30°) = ADJ/HYP = [(√3)/3]/y 

...with the TANGENT and COSINE proportions for the 30°-60°-90° triangle inscribed in the Unit Circle: 
tan(30°) = {(½)/[(√3)/2]} = 1/√3 = [(√3)/3] 
cos(30°) = {[(√3)/2]/(1)} = [(√3)/2] 

Equate them: 
tan(30°) = tan(30°) 
x/[(√3)/3] = 1/√3 
multiply by reciprocals or identity-fractions until you get 
x = ⅓ 

cos(30°) = cos(30°) 
[(√3)/3]/y = [(√3)/2] 
multiply by reciprocals or identity-fractions until you get 
y = 2/3 

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