marigirl one year ago Working on a Differentiation question- need some advice. The Question: A circle has a diameter AB of length 8cm. C is a point on the circumference. Show that the triangle ABC will be isosceles if the perimeter is a maximum and find the maximum perimeter. My solution so far: Since length AB is the diameter, then Angle ACB is 90 degrees. We can use Pythagoras Theorem to express the perimeter of this triangle.

1. marigirl

Length of sides of triangle is a b and 8 $side b= \sqrt{64-a ^{2}}$ Perimeter = 8+ a+b $P= 8+a+\sqrt{64-a ^{2}}$ then I derived this equation $\frac{ dP }{ da }= 1-\frac{ a }{ \sqrt{64-a ^{2}} }$ then made this equation equal to zero to find maximum $a=4\sqrt{2}$ Then Used this Value to find Maximum Perimeter Maximum perimeter value is 19.3cm MY QUESTION: How do I actually write up that I have Triangle ABC is isosceles.?

2. marigirl

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3. anonymous

hmmm just guess.. AC = CB ?

4. marigirl

When I went about calculating the perimeter, I did go about saying side a and side b and side c which is 8cm..... but I am not sure how to mathematically state that what I found is isosceles. (My answer matches up with the model answers provided)

5. anonymous

I see what you mean, Im not sure what the expectation is of your teacher, but I would think if you proved AB <> AC and AC = CB then triangle ABC is isosceles by the isosceles theorem

6. marigirl

oh yes i see it.. i did calculate a and then from that i can get b .. both these lengths are the same .. so i guess that is sufficient enough..