Answer
D(0,0)
E(2a,0)
F(a,7)
Work Step by Step
First, we find the coordinates of D.
From the picture, we see vertex D is located at the origin. Hence the coordinates of D are $(0,0)$.
Next, we find the coordinates of E.
We are given the $x$-coordinate of E, which is $2a$, and from the picture, we see E lies on the $x$-axis, which means the $y$-coordinate of E is $0$. Thus vertex E has coordinates $(2a,0).$
Now we find vertex F. We are given that our triangle is an isosceles triangle where DF=FE. This means F is the vertex angle of this isosceles triangle. So the altitude through vertex F will bisect $\overline{DE}$. Thus the point (let us call it M) where this altitude intersects $\overline{DE}$ is the midpoint of $\overline{DE}.$ Using the midpoint formula, we see the coordinates of M are $\left(\dfrac{2a-0}{2}, \dfrac{0-0}{2}\right)=\left(\dfrac{2a}{2}, \dfrac{0}{2}\right)=(a,2)$. Now, F is located 7 units above M. So F has the same $x$-coordinate as M, which is $a$, and it's $y$-coordinate is 7 more than that of M's $y$-coordinate. Hence the coordinates of F are $(a,7)$.
