SOLUTION: If the lengths of the two equal sides of an isosceles triangle are each x cm, find the length for the third side so that the triangle has maximum area.

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Question 1133761: If the lengths of the two equal sides of an isosceles triangle are each x cm, find the length for the third side so that the triangle has maximum area.
Found 3 solutions by addingup, Alan3354, ikleyn:
Answer by addingup(3677) About Me  (Show Source):
You can put this solution on YOUR website!
Per Pythagoras, third side:
sqrt%28x%5E2+%2B+x%5E2%29

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
If the lengths of the two equal sides of an isosceles triangle are each x cm, find the length for the third side so that the triangle has maximum area.
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The 3rd side's length is 2s.
Area = b*h/2
h+=+sqrt%28x%5E2+-+s%5E2%29
Area+A+=+s%2Asqrt%28x%5E2+-+s%5E2%29+=+%28s%5E2%2Ax%5E2+-+s%5E4%29%5E%281%2F2%29

dA%2Fds+=+%28s%5E2%2Ax%5E2+-+s%5E4%29%5E%28-1%2F2%29%2A%28s%2Ax%5E2+-+2s%5E3%29
Find s for dA/ds = 0
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s%2Ax%5E2+-+2s%5E3+=+0 ---- Ignore s = 0
x%5E2+-+2s%5E2+=+0
s = x*sqrt(2)/2
Side length = x*sqrt(2)
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Might be a lot simpler to assign a number to x, eg, 10.
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The other tutor assumed it's a right triangle, which it is, but he showed no proof of it.
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I would have guessed that the equilateral triangle would give the max area.



Answer by ikleyn(52805) About Me  (Show Source):
You can put this solution on YOUR website!
.
The area of any triangle is

Area = %281%2F2%29%2Aa%2Ab%2Asin%28gamma%29

where a and b are any two sides and gamma is the angle between them.

In given case, the sides are of the length of x centimeters, and the area is maximal when the anle gamma is 90 degrees.

So, the area is maximal when the triangle is isosceles right angled triangle, and then the area is

Area%5Bmax%5D = %281%2F2%29x2.

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Completed and solved.