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# What is the length of the longest side?

http://www.e-zgeometry.com/pow/pictures/16.gif there is the link to the picture. thanks could you add detail on how you solved it? I want to learn how to do it in the future.

Update:

http://www.e-zgeometry.com/pow/pictures/... there is the link to the picture. even tho a side made look the longest, its not the longest. thanks could you add detail on how you solved it? I want to learn how to do it in the future. and also

Update 2:

i mean the pictures is not to scale

Relevance

The longest line is BC. (See the correctly drawn diagram in the attached link.)

This relies on a rule of triangles that the longest side is *always* opposite the biggest angle.

Look at the two upper triangles (ABF and FED). Their biggest angle is on the outside making their longest edges part of the inner triangle (BFD).

However, for that triangle, the longest side is BD which is longer than any of the other sides so far. So far we have eliminated all other sides above as shorter than BD.

That gets us to the bottom triangle whose longest side is BC (longer than BD).

Thus the longest line is BC...

As you noted, the picture is *definitely* not drawn to scale.

Just for fun, I decided to take the measurements and actually draw the image *to scale*. The resulting picture is attached as a link. Hopefully now it is more apparent that BC is the longest side, as proven previously.

In the future, I'd look out for angles like BDC that are deliberately drawn far from their correct value. That was the first clue that BC might be the longest side...

Also, as others have noted, until you are given at least one measurement, you can't determine the actual length, just the relative length. For example, BC is about 4% longer than BD...

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• The longest side *looks* like line segment BD, but since this is just a drawing, the angles could be set up to make it so that the longest side is actually something else.

My recommendation, since there is no length provided for any of the segments, is to choose a value (100 would be convenient) and assign it to an arbitrary side. Then, compute the lengths of any other side by trigonometry. The sine rule will be helpful here.

for example, if we make length BD 100, and angle FBD is 50, 100 / sin 100 = FB / sin 50. This will simplify to FB = (100 x sin 50) / sin 100, which is about 77.77, and the other sides can be figured out the same way.

Once you have more than one side known, you can use the cosine rule as well.

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• hayharbr is right and YES, the figure is seriously misdrawn. (Note that the 70 deg. angle in the lowest triangle is MUCH SMALLER than the 45 deg. angle!)

What's missing from all the above is the actual PRINCIPLE involved. I think THAT'S what you're asking for ("THE SINE RULE").

There's a trig. relationship:

a/sin A = b/sin B = c/sin C where the angle A is opposite the side a, etc., in a GIVEN triangle.

THAT'S why the LARGEST LENGTH is OPPOSITE the LARGEST ANGLE in any GIVEN triangle. (Obvious when all angles are acute; and with a little more thought, also when one is obtuse: each OTHER angle is then necessarily SMALLER than the difference of 180 deg.and the largest angle, so their sines are still smaller.) So, it's by using this principle that you can end up with the chain of argument already given.

Live long and prosper.

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• You can't find its length but you can find which side it is.

The longest side is across from the largest angle in each triangle. The longest side of triangle ABF is FB but that's shorter than BD in the center triangle. The longest side of triangle FED is FD but that's also shorter than BD in the center triangle. However, in triangle BCD, BC is the longest side, even longer than BD. So BC is the longest side of all.

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• The longest side of a triangle is always opposite to the largest angle. All of the outside triangle's longest sides are the sides of the inside triangle and the longest side of the inside triangle is BD, therefore the longest side is BD.

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• it's impossible to find the length of any of those sides

the above guy was right the side opposite the largest angle is the largest side (therefore the side oppposite the angle of 100 degrees is the largest)

but without a length of any of the sides of the triangles, you cant find a length of the longest side, this is because triginometry requires at least 1 side, and 90 degree angle.

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• use trig and figure it out

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• not enough info to do it.

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