Geometry
# Area of Triangles

Triangle $ABC$ has coodinates $A= (-4, 0)$, $B= (4 , 0)$, and $C= (0 , 3)$.

Let $P$ be the point in the first quadrant such that $\triangle ABP$ has half the area of $\triangle ABC$ but both triangles have the same perimeter.

What is the length of $CP?$ If your solution is in a form of $\sqrt{d}$, submit $d$ as the answer.

Let $ABCD$ be a square of side length 12.

- $E$ is the midpoint of $CB$,
- $FC = \frac{1}{3} DC$,
- $GD = \frac{1}{4} DA$,
- $AH = \frac{1}{3} AE$,
- $J$ is the midpoint of $FE$.

What is the area of the purple triangle?

Which of the following triangles has a larger area:

- triangle A with side lengths $13, 13, 10$, or
- triangle B with side lengths $13, 13, 24\, ?$

What is the largest possible area of an isosceles triangle with two sides of length 2?