Problem 5 · Two circles in a rectangle
Pythagoras between centres: distance $13$, legs $5$ and $12$.
Integer answer, at most 4 digitsThe figure shows two circles of diameters $10$ and $16$ centimetres inside a rectangle with horizontal base $18$ centimetres, so that the circles are tangent to each other and to the rectangle. What is the height of the rectangle?
Reasoned solution
Key idea: two externally tangent circles have centres at distance $r_1 + r_2$; the tangencies with the rectangle pin down the centres' coordinates.
Radii $5$ and $8$. The small circle touches the top and left sides: centre $(5,\, H-5)$. The big one touches the bottom and right sides: centre $(18-8,\, 8) = (10,\, 8)$. The distance between centres is $5 + 8 = 13$:
(The $5$–$12$–$13$ triangle: horizontal gap $5$, vertical $12$.) The height is $5 + 12 + 8 = 25$ cm.