SOLUTION: Hi, i'm having a bit of trouble do you think you could help? the angles of a triangle are in the ratio 3:4:5. If the shortest side has a length of 1 meter, what idsthe length

Algebra ->  Triangles -> SOLUTION: Hi, i'm having a bit of trouble do you think you could help? the angles of a triangle are in the ratio 3:4:5. If the shortest side has a length of 1 meter, what idsthe length       Log On


   

Question 168863: Hi,
i'm having a bit of trouble do you think you could help?
the angles of a triangle are in the ratio 3:4:5. If the shortest side has a length of 1 meter, what idsthe length of the longest side?
thanks!
=)
Answer by jojo14344(1513) About Me  (Show Source):
You can put this solution on YOUR website!
.
To begin with, 1st we need to find the ANGLES of the triangle.
Given Ratio ----> 3:4:5 ------>system%28A=45%5Eo%2CB=60%5Eo%2C%2BC=75%5Eo%2C180%5E0%29
Agree? Divide All Angles by 15: 45%5Eo%2F15=highlight%283%29;60%5E0%2F15=highlight%284%29;75%5Eo%2F15=highlight%285%29
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Let's put those angles in figure so it will be easy to analyze:
---> As you see in our drawing,
BC is the shortest side = 1meter.
For argument sake, take a protractor and
draw on your own to have the right angles and measure the shortest.
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Here's the trick, determining the longset side because it seems the other 2 sides are so close. Draw a Line from B to Line AC and mark point D:

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We can solve for Line BD by Pyth Theorem -->%28BC%29%5E2=%28BD%5E2%29%2B%28DC%29%5E2 -->
%28BD%29%5E2=%28BC%29%5E2-%28DC%29%5E2 ---------> EQN 1
*note: BC=1m, and we need to find DC?
By trigo function: Cos75%5Eo=adj%2Fhyp=DC%2FBC--->DC=%28cos75%5Eo%29%28BC%29=%28cos75%5Eo%29%281%29 --> DC=0.2588m, Plug in EQN 1:
%28BD%29%5E2=1%5E2-0.2588%5E2=1-0.066977=%280.93302m%29
BD=sqrt%280.93302%29=0.966023m
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Now we need to find AD. By Trigo function, Tan45%5Eo=BD%2FAD
AD=BD%2FTan45%5Eo=0.966023%2FTan45%5Eo=0.966023m
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So, line AC=AD%2BDC=0.966023%2B0.2588=1.2248m
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Now,we find Line AB. We can do that either by Trigo Function or Cosine Law. Let's try both and we'll see if the same.
By Trigo function---> Sin45%5Eo=opp%2Fhyp=BD%2FAB
AB=BD%2FSin45%5Eo=0.966023%2FSin45%5Eo=1.36616m
Or By Cosine Law ---> %28AB%29%5E2=%28AC%29%5E2%2B%28BC%29%5E2-2%28AC%29%28BC%29Cos75%5Eo
%28AB%29%5E2=1.2248%5E2%2B1%5E2-2%281.2248%29%281%29Cos75%5Eo
%28AB%29%5E2=1.5%2B1-0.6340=1.866
AB=sqrt%281.866%29=1.366m, The same above
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So here are your measurements:
highlight%28AB=1.366m%29 ----------> Longest
BC=1m -------------------------> Shortest
AC=1.2248m --------------------> Middle

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Thank you,
Jojo