The parabola with equation $y=25-x^{2}$ intersects the $x$-axis at points $A$ and $B$, as shown.

(a) Determine the length of $A B$.

(b) Rectangle $A B C D$ is formed as shown with $C$ and $D$ below the $x$-axis and $B D=26$. Determine the length of $B C$.

(c) If $C D$ is extended in both directions, it meets the parabola at points $E$ and $F$. Determine the length of $E F$.

![](https://cdn.mathpix.com/cropped/2024_04_17_7e8f36a7970c2b706527g-3.jpg?height=548&width=450&top_left_y=474&top_left_x=1268)
(a) The points $A$ and $B$ are the points where the parabola with equation $y=25-x^{2}$ intersects the $x$-axis.

To find their coordinates, we solve the equation $0=25-x^{2}$ to get $x^{2}=25$ or $x= \pm 5$.

Thus, $A$ has coordinates $(-5,0)$ and $B$ has coordinates $(5,0)$.

Therefore, $A B=5-(-5)=10$.

(b) Since $A B C D$ is a rectangle, $B C=A D$ and $\angle D A B=90^{\circ}$.

Since $B D=26$ and $A B=10$, then by the Pythagorean Theorem,

$$
A D=\sqrt{B D^{2}-A B^{2}}=\sqrt{26^{2}-10^{2}}=\sqrt{676-100}=\sqrt{576}=24
$$

since $A D>0$.

Since $B C=A D$, then $B C=24$.

(c) Since $A B C D$ is a rectangle with sides parallel to the axes, then $D$ and $C$ are vertically below $A$ and $B$, respectively.

Since $A D=B C=24, A$ has coordinates $(-5,0)$ and $B$ has coordinates $(5,0)$, then $D$ has coordinates $(-5,-24)$ and $C$ has coordinates $(5,-24)$.

Thus, line segment $D C$ lies along the line with equation $y=-24$.

Therefore, the points $E$ and $F$ are the points of intersection of the line $y=-24$ with the parabola with equation $y=25-x^{2}$.

To find their coordinates, we solve $-24=25-x^{2}$ to get $x^{2}=49$ or $x= \pm 7$.

Thus, $E$ and $F$ have coordinates $(-7,-24)$ and $(7,-24)$ and so $E F=7-(-7)=14$.