Multiple choice

Find the maximum perimeter of a triangle on a given base '$a$' and having the given vertical angle $\displaystyle \alpha$

  1. $\displaystyle P_{max}=a\left ( 1+\mathrm{cosec}\frac{\alpha }{2} \right )$
  2. $\displaystyle P_{max}=2a\left ( 1+\cos\frac{\alpha }{2} \right )$
  3. $\displaystyle P_{max}=2a\left ( 1+\mathrm{cosec}\frac{\alpha }{2} \right )$
  4. $\displaystyle P_{max}=a\left ( 1+\cos\frac{\alpha }{2} \right )$
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A Correct answer
Explanation

The maximum perimeter of a triangle with a fixed base and vertical angle is achieved when the triangle is isosceles.

AI explanation

Using the properties of a triangle, if the base is a and the vertical angle is alpha, the maximum perimeter occurs when the triangle is isosceles. By the sine rule, the equal sides are found by splitting the triangle into two right triangles, yielding side lengths of a/(2 sin(alpha/2)). The maximum perimeter Pmax is the sum of the base and the two equal sides, Pmax = a + 2(a/(2 sin(alpha/2))), which simplifies to Pmax = a(1 + cosec(alpha/2)).

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