Right-angled triangle

Since $\angle ACB=90°$ , the $CXPY$ quadrilateral is a rectangle, so $XY=CP$ . So we need to find the minimum value of $CP$ . It's clear that it becomes the least when $\angle CPA=90°$ , ie $CP$ is a perpendicular. So imagine that now $\overline{XY}$ is the least. Here the area of the triangle is: $\begin{aligned} \overline{AC}*\overline{BC} & = 3*4 \\ & = 12 \\ & = \overline{CP}*\overline{AB} \\ & = \sqrt{3^2+4^2}*\overline{CP} \\ & = 5\overline{CP} \end{aligned}$ From that $\text{min}(\overline{CP})=\dfrac{12}{5}$ .

Therefore the answer is $\boxed{17}$ .

Let $\overline{CX} = x$ and $\overline{PX} = y$ . Then, we have:

$\begin{aligned} \overline{XY} & = \sqrt{x^2+{\color{#3D99F6}y^2}} & \small \color{#3D99F6} \text{Note that }y = \frac {4(3-x)}3 \\ & = \sqrt{x^2+{\color{#3D99F6}\frac {16}9(3-x)^2}} \\ & = \sqrt{\frac{25x^2-96x+144}9} \\ & = \sqrt{\frac{25}9 \left(x^2-\frac {96}{25}x + \left(\frac {96}{50}\right)^2 - \left(\frac {96}{50}\right)^2 +\frac {144}{25} \right)} \\ & = \sqrt{\frac{25}9 \left({\color{#3D99F6}\left(x-\frac {96}{50}\right)^2} +\frac {1296}{25^2} \right)} & \small \color{#3D99F6} \text{Note that }\left(x-\frac {96}{50}\right)^2 \ge 0 \end{aligned}$

$\begin{aligned} \implies \min \left(\overline{XY}\right) & = \sqrt{\frac{25}9 \left({\color{#3D99F6}0} +\frac {1296}{25^2} \right)} = \frac {12}5 \end{aligned}$

$\implies m+n = 12+5 = \boxed{17}$

Let $XC=a$ , $YC=XP=b$ hence $XB=4-a$ Now observe XBP is similar to CBA so we have $\frac{4-a}{b}=\frac{4}{3} \to 12=4b+3a$ . Now we wish to minimze $\sqrt{a^2+b^2}$ which is equivalent to minimizing the distance from the origin to the line $12=4b+3a$ . Hence we take the perpendicular distance which is trivially $\frac{12}{5}$ $\boxed{17}$

Clearly, YX = CP. We need to locate point P on the hypotenuse so that CP is perpendicular to BA. The slope of BA is -4/3. we want the slope of CP to be 3/4, so that the product of slopes = -1. Then if (x,y) are the coordinates of P, we have ,y = (3/4)x, and (CP)^2 = x^2 + (3x/4)^2, so CP = (5/4)x. We have: (3/4)x/(3 - x) = 4/3, or x = 48/25. then CP = (5/4)*(48/25) = 12/5, and the sum = 17. Ed Gray