Math CentralQuandaries & Queries


Alannah, a student:

I have a word problem from my Calculus textbook that I can't figure out.

Triangle ABC is inscribed in a semicircle with diameter BC=10cm. Find the value of angle B that produces the triangle of maximum area.

I am supposed to set up an equation for the area of the triangle A=b x h/2 using Trig functions based on angle B to represent the base and height but I'm not sure how to do this when the side length given is not the hypotenuse.

Hi Alannah,

Any triangle inscribed in a semi-circle, will have a right angle. If either angle B or C are 90o then the triangle would not be inscribed:

That means that angle A is 90o so BC must be the hypotenuse. We can use trig ratios to find length of the height and base of the triangle.

cosB = adjacent side to B/ hypotenuse = AB/10 ⇒ AB = 10 cos B

sinB = opposite side to B/ hypotenuse = AC/10 ⇒ AC = 10 sin B

The formula for the area with respect to angle B

Area = 0.5bh=50 cosB sinB

Hope this helps,


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