Expert: Sherman D. - 4/9/2008

Question
Find the interior angles of the triangle with the given vertices.
(-4,5), (1,10), (3,1)
I really don't understand how we're supposed to do this.  I've tried everything!

Answer
just use the distance formula this way you'll know about what triangle we are dealing with. By find out what your triangle is, you will know rather to use Pythagorean Thereom or Law of Cosine

D = sqrt((x2 - x1)^2 + (y2 - y1)^2)

(-4,5) and (1,10)
D = sqrt((1 - (-4))^2 + (10 - 5)^2)
D = sqrt((1 + 4)^2 + 5^2)
D = sqrt(5^2 + 5^2)
D = sqrt(50)
D = 5sqrt(2)

(1,10) and (3,1)
D = sqrt((3 - 1)^2 + (1 - 10)^2)
D = sqrt(2^2 + (-9)^2)
D = sqrt(4 + 81)
D = sqrt(85)

(-4,5) and (3,1)
D = sqrt((3 - (-4))^2) + (1 - 5)^2)
D = sqrt((3 + 4)^2 + (-4)^2)
D = sqrt(7^2 + 16)
D = sqrt(49 + 16)
D = sqrt(65)

as you can see we have a scalene triangle, so you will have to use Law of Cosines, also you will notice that the a^2 + b^2 = c^2 doesn't work out, so its not a scalene right triangle.

a^2 = b^2 + c^2 - 2bc(cos(A))

when you do the sqrt(A)^2, the sqrt and ^2 cancel out

50 = 85 + 65 - 2(sqrt(85) * sqrt(65))cos(A)
50 = 150 - 2sqrt(5525)cos(A)
-100 = -2sqrt(25 * 221)cos(A)
-100 = -10sqrt(221)cos(A)
cos(A) = 10/sqrt(221)
A = cos^-1(10/sqrt(221))
A = 47°43'24.72" or about 47.7263°

85 = 50 + 65 - 2(sqrt(50) * sqrt(65))cos(B)
85 = 115 - 2sqrt(3250)cos(B)
-30 = -2sqrt(25 * 130)cos(B)
-30 = -10sqrt(130)cos(B)
cos(B) = 3/sqrt(130)
B = cos^-1(3/sqrt(130))
B = 74°44'41.57 or about 74.7449°

65 = 50 + 85 - 2(sqrt(50) * sqrt(85))cos(C)
65 = 135 - 2sqrt(4250)cos(C)
-70 = -10sqrt(170)cos(C)
cos(C) = 7/sqrt(170)
C = cos^-1(7/sqrt(170))
C = 57°31'43.71" or about 57.5288

ANS:
A = 47°43'24.72" or about 47.7263°
B = 74°44'41.57 or about 74.7449°
C = 57°31'43.71" or about 57.5288

if you add the all angles together, you will get 180, which proves that all the angles are correct, because all the interior angles of a triangle add up to be 180.