There's a circle of diameter $d$ that is on a wall, and touches a block. Find the value of $d$.
I'm really unsure how to go about solving this. I wanted to first approach this by using the arc length, but I'm really unsure how to proceed. Anyone have ideas?
If appears that the circle is touching a surface to its left and another surface below it.
If we were to graph this circle with these two surfaces representing the $x$ and $y$ axes, the circle would have the equation
$$(x-r)^2+(y-r)^2=r^2$$
in order for it to just touch either surface.
If the circle also just touches the block that is $5$ units wide and $8$ units long, the point $(5,8)$ must be on the circle. If we plug this value into the equation, we get
$$(5-r)^2+(8-r)^2=r^2$$
$$25-10r+r^2+64-16r+r^2=r^2$$
$$r^2-26r+89=0$$
If we use the quadratic formula, we get solutions of
$$r=13\pm4\sqrt{5}$$
One of these solutions, $13-4\sqrt{5}$, doesn't work, though, because the radius becomes too small to leave enough room in the bottom left corner for the block.
This means that
$$r=13+4\sqrt{5}$$
which consequently means
$$d=2r=2(13+4\sqrt{5})=26+8\sqrt{5}$$