# A wagon wheel is placed against a wall. One point on the edge of the wheel is 5 inches from the ground and 10 inches from the wall. What is the radius of the wheel?

Nov 19, 2016

$r = 25$inches

#### Explanation:

Acknowledgements to Stefan. V for this solution.

The trick here is the placement of the wheel.
Note that the circumference touches both the wall and the ground.

Check out this diagram

So if you start with the equation of a circle with the centre at
$\left(h , k\right) \mathmr{and} \text{ radius } r$, then ${\left(h - x\right)}^{2} + {\left(k - y\right)}^{2} = {r}^{2}$

Use the fact that the wheel is touching both the wall and the ground at a distance equal to its radius from the centre to say that

$h = r \mathmr{and} k = r$

This means that for any point $\left(x , y\right)$ on the circumference, you can draw a right-angled triangle inside the circle, which will have sides of length: $\left(r - x\right) , \left(r - y\right) \mathmr{and} r$ as the hypotenuse.

This gives:

${\left(r - x\right)}^{2} + {\left(r - y\right)}^{2} = {r}^{2}$

Now it's just a matter of plugging in the point $\left(10 , 5\right)$

${\left(r - 10\right)}^{2} + {\left(r - 5\right)}^{2} = {r}^{2}$

${r}^{2} - 20 r + 100 + {r}^{2} - 10 r + 25 = {r}^{2}$

${r}^{2} - 30 r + 125 = 0$

$\left(r - 25\right) \left(r - 5\right) = 0$

$x = 25 \mathmr{and} x = 5$
$r = 25$