Thread: College geometry: Minumum and triangle problem

Show that for a triangle with given area and given side, the sum of the other two sides is a minimum if and only if the triangle is isosceles.

Thanks for your help.

Originally Posted by ordinalhigh

Show that for a triangle with given area and given side, the sum of the other two sides is a minimum if and only if the triangle is isosceles.

Thanks for your help.

Draw a parallel line from the vertex to the given side.
Taking any point on this lin, if you draw a triangle with the given side, all the triangles will have the same area.
If a and b are the sides of the triangles,and b> a then angle B>A.
According to sine rule
a/sinA = b/sinB.
So a = k*sinA
b = k*sinB.
Hence a + b = k(sinA + sinB) > 2*k*sinA
So a+b will be minimum if a = b.