In right triangle $ABC$, $BD=CD+9$. If $AB=10$ and $BC=25$, what is $AD$
Let $AC=y$. We note that $AD=AC-CD,$ so we need to find $AC$ and $CD$ separately. By Pythagoras on triangle $ABC,$ $AC=\sqrt{25^2-10^2}=15\sqrt{3}.$ Next, notice that $\triangle CDA$ and $\triangle BDA$ share hypotenuse $AD.$ Using the $HL$ congruency criterion, it follows that $\triangle CAD\cong \triangle BAD.$ In particular, $CD=DA=AC-CD.$ We then solve for $CD$ as follows.
\begin{align*} CD&=DA=AC-CD \\ \Rightarrow CD+CD&=15\sqrt{3} \\ \Rightarrow 2\cdot CD&=15\sqrt{3} \\ \Rightarrow CD&=\frac{15\sqrt{3}}{2}. \\ \end{align*}
Finally, \[AD=AC-CD=15\sqrt{3}-\frac{15\sqrt{3}}{2}=\frac{15\sqrt{3}}{2}.\]$\boxed{\frac{15\sqrt{3}}{2}}$
To find $AD$, we can use the Pythagorean Theorem for right triangle $ABC$.
The Pythagorean Theorem states that for any right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.
In this case, we have $AB = 10$ and $BC = 25$. Let $AC$ be the hypotenuse.
The Pythagorean Theorem can be written as: $AB^2 + BC^2 = AC^2$
Plugging in the given values, we have $10^2 + 25^2 = AC^2$
Simplifying, we get $100 + 625 = AC^2$
Combining like terms, we have $725 = AC^2$
To find $AC$, we take the square root of both sides: $\sqrt{725} = AC$
Calculating the square root of 725, we get $26.91 \approx AC$
Now, let's find the length of $BD$.
We are given that $BD = CD+9$
Let's call $CD$ as $x$.
Therefore, $BD = x + 9$
Now, we have two equations:
1. $AB = 10$
2. $BD = x + 9$
We know that $AB = AC - CD$ (since $AB = AC$ and $CD = 0$)
Substituting the values, we get $10 = 26.91 - x$
Solving for $x$, we get $x = 26.91 - 10 = 16.91$
Therefore, $BD = x + 9 = 16.91 + 9 = 25.91 \approx 26$
Now, we can find $AD$ by using the Pythagorean Theorem for right triangle $ABD$.
$AB^2 + BD^2 = AD^2$
Substituting the values, we have $10^2 + 26^2 = AD^2$
Simplifying, we get $100 + 676 = AD^2$
Combining like terms, we have $776 = AD^2$
To find $AD$, we take the square root of both sides: $\sqrt{776} = AD$
Calculating the square root of 776, we get $27.87 \approx AD$
Therefore, $AD \approx 27.87$