Question

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.

Answer 1

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.?

Answer 2

hmmm just guess.. AC = CB ?

Answer 3

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)

Answer 4

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

Answer 5

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..