SOLUTION: A triangle has coordinates of (1,1), (4,1), and (3,4). is it a 30-60-90 triangle and why?

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Question 959109: A triangle has coordinates of (1,1), (4,1), and (3,4). is it a 30-60-90 triangle and why?
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!
THERE ARE TWO special triangles in trigonometry. One is the 30°-60°-90° triangle. The other is the isosceles+right+triangle. They are special because, with simple geometry, we can know the ratios of their sides.

so, what we need to do first is find the sides length using given points and distance formula
(1,1), (4,1), and (3,4)
the distance between (1,1)and (4,1)
d=sqrt%28%284-1%29%5E2%2B%281-1%29%5E2%29
d=sqrt%283%5E2%2B0%5E2%29
d=sqrt%283%5E2%29
d=3

the distance between (1,1)and (3,4)
d=sqrt%28%283-1%29%5E2%2B%284-1%29%5E2%29
d=sqrt%282%5E2%2B3%5E2%29
d=sqrt%2813%29-exactly
d=3.6-approximately

the distance between (4,1) and (3,4)
d=sqrt%28%284-3%29%5E2%2B%281-4%29%5E2%29
d=sqrt%281%5E2%2B%28-3%29%5E2%29
d=sqrt%2810%29-exactly
d=3.2-approximately
so, the shortest side is 3 units long, then the longer side is sqrt%2810%29 and the longest side is sqrt%2813%29

now, check if these are sides of a 30°-60°-90° triangle
first, use Pythagorean theorem because triangle has an 90° angle
%28sqrt%2813%29%29%5E2=%28sqrt%2810%29%29%5E2%2B3%5E2
13=10%2B9
13%3C%3E19=>your triangle is not a 30°-60°-90° triangle
other way is to use a theorem:
In a 30°-60°-90° triangle the sides are in the ratio 1+%3A+2+%3A+sqrt%28+3%29.
now, the sides of this triangle are in ratio 3%3Asqrt%2810%29%3Asqrt%2813%29
compared to 1+%3A+2+%3A+sqrt%28+3%29, we can see that these ratios are not equivalent; so, your triangle is not a
30°-60°-90° triangle

now, see it on a graph: