# Thread: triangles and trig and all that

1. ## triangles and trig and all that

I need to get the length of the red line (height of the triangle), and the length of the black and the green line too

The green and black lines combined are 500

Also, the entire triangle won't necessary be a right-angled one

Thanks to anyone who can help, I'm brain dead at the moment

2. posted it in galah but its 240

3. that's good thanks but how did you get that...i need to be able to do it for any length of sides

4. write the black line as x, so the green line is (500-x). let the red line be k. set up two pythagorus equations. solve for x. then solve for k.

5. ## right angled

Another possible way using areas:
1) Work out the angle opposite the side 300 (use cosine rule)
2) Work out the area using this angle and the sine rule
3) Divid the area by 2, then by 500, and you get the height

i.e. cos (A) = (400^2 + 500^2 - 300^2)/(2*400*500)
=> work out A
=> know sin(A) = (1- cos(A)^2) ^ 1/2
=> know area = 1/2 * 400 * 500 * sin(A)
=> know height = area/(2*500)
=> work out green line using cosine rule in 400-red line-green line triangle
=> work out black line = 500-green line

6. The entire triangle must be right angled, because it obeys pythagoras theorem.

Try two simultaneous equations with the sine rule (calling the red line R):

Call black line X and red line R, angles in degrees for ease of use:

sin60 / R = sin30 / X
X/2 = R(Root3/2)
X = R(Root3)

sin30 / R = sin60 / (500 - X)

Sticking in for X, that gets you:
1 / 2R = (Root3/2) / (500 - R(Root3))

So R = 250 / Root3 = 144.33756729740644112728719512549...

Now that's assuming those angles are what I educatedly guessed them to be. I mean it all looks nice and neat that way.

7. And for my next amazing feat of genius, I will disappear from these forums for an entire week because I'm going skiing in France! Cyas!

8. Use Heron's Formula; it has never failed since I used it to solve the first difficult geometry problem that I could do.

9. with Assbandit noticing that A+B=90, and you know the side lengths, use the law of tangents!

10. ## Fastest

The fastese way to do it on that diagram is, knowing that it is a right-angled triangle, is say that redline*500 = 300*400 (= 2*area)
=> redline = 300*400/500
= 3*400/5 = 3*80 = 240
Green/Red = Red/Black
=> 240^2 = Green(500-Green) = 500 Green - Green^2
=> green^2 - 500 green + 240^2 = 0
(green - 180)(green-320) = 0
=> green = 320, black = 180 (green>black)

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