SOLUTION: What are the angle measure of the triangle if the measures are 21, 7 sqrt 3, and 14 sqrt 3?

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Question 1152046: What are the angle measure of the triangle if the measures are 21, 7 sqrt 3, and 14 sqrt 3?
Found 3 solutions by Boreal, MathLover1, ikleyn:
Answer by Boreal(15235) About Me  (Show Source):
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My approach to this is to note that 7 sqrt(3) is half of 14 sqrt (3), and this is probably going to be a 1-2-sqrt(3) 30-90-60 right triangle
Sure enough, 7 sqrt(3) *sqrt(3)=21
30-60-90 degree right triangle.

Answer by MathLover1(20850) About Me  (Show Source):
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What are the angle measure of the triangle if the measures are
a=21
b=7sqrt%283%29=12.12
c=14sqrt%283%29=24.25


Angle ∠ A+=cos%5E-1%28%28b%5E2%2Bc%5E2-a%5E2%29%2F2bc%29

Angle ∠ A+=cos%5E-1%28%2812.12%5E2%2B24.25%5E2-21%5E2%29%2F%282%2A12.12%2A24.25%29%29

Angle ∠ A+=cos%5E-1%28293.96%2F587.82%29

Angle ∠ A+=1.04711 rad
Angle ∠ A+=59.995° = 59°59'41"....round it
Angle ∠ A+=60°


Angle ∠ B+=+cos%5E-1%28%28a%5E2%2Bc%5E2-b%5E2%29%2F2ac%29

Angle ∠ B++=cos%5E-1%28%2821%5E2%2B24.25%5E2-12.12%5E2%29%2F%282%2A21%2A24.25%29%29

Angle ∠ B++=cos%5E-1%28882.17%2F1018.5%29

Angle ∠ B++=0.52336 rad
Angle ∠ B++=29.986° = 29°59'11"......round it
Angle ∠ B++=30°

Angle ∠ C+=cos%5E-1%28%28a%5E2%2Bb%5E2-c%5E2%29%2F2ab%29

Angle ∠ C+=cos%5E-1%28%2821%5E2%2B12.12%5E2-24.25%5E2%29%2F%282%2A21%2A12.12%29%29

Angle ∠ C+=cos%5E-1%28-0.1681%2F509.04%29

Angle ∠ C+=1.57113 rad
Angle ∠ C+=+90.019° = 90°1'8"......round it

Angle ∠ C+=+90°




Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.
What are the angle measures of the triangle? the highlight%28cross%28angles%29%29 sides are 21, 7 sqrt 3 and 14 sqrt 3
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            Please notice on how I  edited  your  post  in order for to make sense.


Solution 


Notice that



    %287%2Asqrt%283%29%29%5E2 + 21%5E2 = 49*3 + 441 = 147 + 441 = 588= 3*196 = 14%5E2%2A%28sqrt%283%29%29%5E2 = %2814%2Asqrt%283%29%29%5E2.



Thus the triangle with the side lengths of 21, 7%2Asqrt%283%29  and  14%2Asqrt%283%29  is a right angled triangle

with the legs  21  and  7%2Asqrt%283%29  and the hypotenuse of  14%2Asqrt%283%29.


Also,  the leg  7%2Asqrt%283%29   is half length of the hypotenuse  14%2Asqrt%283%29 .


Therefore, the given triangle is  (30°, 60°, 90°)- triangle.


Its angles are 30°,  60°  and  90°.

Solved.