Question

perimeter of the isosceles triangle​

Answer 1

Explanation:

The perimeter {\displaystyle p} of an isosceles triangle with equal sides {\displaystyle a} and base {\displaystyle b} is just

{\displaystyle p=2a+b.}

As in any triangle, the area {\displaystyle T} and perimeter {\displaystyle p} are related by the isoperimetric inequality

{\displaystyle p^{2}>12{\sqrt {3}}T.}

This is a strict inequality for isosceles triangles with sides unequal to the base, and becomes an equality for the equilateral triangle. The area, perimeter, and base can also be related to each other by the equation

{\displaystyle 2pb^{3}-p^{2}b^{2}+16T^{2}=0.}

If the base and perimeter are fixed, then this formula determines the area of the resulting isosceles triangle, which is the maximum possible among all triangles with the same base and perimeter. On the other hand, if the area and perimeter are fixed, this formula can be used to recover the base length, but not uniquely: there are in general two distinct isosceles triangles with given area {\displaystyle T} and perimeter {\displaystyle p} . When the isoperimetric inequality becomes an equality, there is only one such triangle, which is equilateral.

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