Triangle ABC has vertices (4,-1), B(5,6) and C(1,3). Prove using coordinate geometry that ABC is an isosceles right triangle.
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<h3>Answer:</h3><h3>C = 28.2, A = 25.8, a = 6.5</h3>
See diagram below.
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Work Shown:
Given info is
B = 126 degrees
b = 12
c = 7
Use the Law of Sines to solve for angle C
sin(C)/c = sin(B)/b
sin(C)/7 = sin(126)/12
sin(C)/7 = 0.067418082864579
sin(C) = 7*0.067418082864579
sin(C) = 0.471926580052053
C = arcsin(0.471926580052053) or C = 180-arcsin(0.471926580052053)
C = 28.1594278560921 or C = 180-28.1594278560921
C = 28.1594278560921 or C = 151.840572143908
C = 28.2 or C = 151.8
We have two possible angle values for C.
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If C = 28.2, then A = 180-B-C = 180-126-28.2 = 25.8
If C = 151.8, then A = 180-B-C = 180-126-151.8 = -97.8
So it is not possible for C = 151.8 (because it leads to angle A being negative)
Therefore, only C = 28.2 is possible.
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Use the law of cosines to find the remaining side 'a'
a^2 = b^2 + c^2 - 2*b*c*cos(A)
a^2 = (12)^2 + (7)^2 - 2*(12)*(7)*cos(25.8)
a^2 = 144 + 49 - 168*0.900318771402194
a^2 = 144 + 49 - 151.253553595569
a^2 = 41.7464464044315
a = sqrt(41.7464464044315)
a = 6.46114900032738
a = 6.5
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Only one triangle is possible
The fully solved triangle has these angles and sides:
- A = 25.8 (approx)
- B = 126
- C = 28.2 (approx)
- a = 6.5 (approx)
- b = 12
- c = 7
With stuff in bold representing the terms we solved for previously. Attached below is an image of the fully solved triangle.