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Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius $ r $.

$$\sqrt{3} r$$

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University of Michigan - Ann Arbor

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Yes, we'll have to find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius art. She's got to stop Nice little answers. So listen to draw a quick figure, we have our circle of radius R. Sure, and then we have my sausages triangle inside of this. I'm gonna draw this in a particular way. Love is in the Texas Chainsaw Army, so these two sides will say have the same length The Texas Chainsaw remake. It wasn't bad. People hate it because there's no reason to remake the original which, and we'll say that this side as a length of B, the base and a distance, um, full metal jacket, the center to each of these corners are and the and we'll call this distance from the center to the base is X. Now, from the figure our Pythagorean theorem, it's clear that half of the base be over to is equal to the square root of R squared minus X squared. Therefore, based B is equal to two times the square root of our sport. Nice experts. We have the base in terms of x Bye. Wow, I told Evan Williams that Caroline's on Broadway was in yield Dying now the area of a triangle A. This is one half times the base times the height. I already have a base in terms of X. What is the heightened inspects If you look at the figure the height is going to be are what? Sex? Yeah. So this is one half times I'm sitting two times the steroid of R squared minus expert times X plus. It is equal to X plus eight times the square root of R squared minus X. Where now, in order to find the triangles, the largest area, and then take the derivative of A with respect to X these experts are not, experts say, and the white with each what was right makes excellent. Yes, this is I've been I've been back to myself one times the square root of r squared minus X squared plus x plus r times. This is one half times, uh, one over the square root of our square glass. X squared times negative two x This is minus exports are times X over the square root of X squared money, the iceberg and texture. We want to find when this is equal to zero. Mm Can't pay us. Wait, How about this bitch They can be And stopping the killing You can You can suck are now finally hard So we have the square root of r squared Minus X squared equals X plus r times X over squared of r squared minus X squared So we have our squared minus X squared is equal to X squared What's Rx Now? This is two x square plus r x minus r squared equals zero Yeah, yeah, And we can actually use the quadratic formula for this factor directly. Think back then directly is the easiest factor This urge to x Yeah, minus r times x plus r equals zero soon as soon as thick skin touches latex And so we get that X equals are over 24 X equals negative are now very clearly X has to be positive. So excuse are over two is our only critical value from shooting are critical value into our equation. We get the base he Well, this is two times swear word of r squared minus are over. Two squared two times the square root of R squared minus R squared over four, which is three R squared over four, which is to over two is one times Route three are really So this is our base and the heights h is exports are which is our over to us are or three are over to You're talking about to eat. So the bases Route three are their stuff have doesn't and the height is three halves are anyway was