# Thread: Find the largest possible area of a triangle inscribed in a semicircle

1. ## Find the largest possible area of a triangle inscribed in a semicircle

A right-angled triangle ABC is inscribed in a semi-circle of radius 1cm, with one of its non-hypotenuse side AB lying on the diameter. Find the largest possible area of triangle ABC.

2. ## Re: Find the largest possible area of a triangle inscribed in a semicircle

$\displaystyle A = \frac{1}{2}bh$

the base of the triangle will be the diameter ... at what point on the semicircle will the triangle have the greatest height?

3. ## Re: Find the largest possible area of a triangle inscribed in a semicircle

Umm Skeeter I think you have misinterpreted the problem. The non hypotenuse side of the triangle is on the diameter, but it cannot be the diameter, or else the triangle will not be right-angled.

4. ## Re: Find the largest possible area of a triangle inscribed in a semicircle

Originally Posted by shiny718
Umm Skeeter I think you have misinterpreted the problem. The non hypotenuse side of the triangle is on the diameter, but it cannot be the diameter, or else the triangle will not be right-angled.
my mistake ...

consider the graph of the semicircle of radius 1 centered at the origin

$\displaystyle y = \sqrt{1 - x^2}$

base of triangle = $\displaystyle 1 + x$

height of triangle = $\displaystyle y = \sqrt{1 - x^2}$

$\displaystyle A = \frac{1}{2}(1+x)\sqrt{1-x^2}$

Not knowing what your background is, either graph the area function in your calculator to find the maximum, or use calculus to optimize the area.

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### maximum area of triangle lying in a semicircle of radius r

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