You may also like

problem icon

Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

problem icon

Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

problem icon

Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

One or Two

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2
Let the radius of each semicircle be $r$.
In the top diagram, let the side-length of the square be $2x$. By Pythagoras' Theorem, 
$x^2 + (2x)^2 = r^2$ and so $5x^2 = r^2$. So this shaded area is $(2x)^2 = 4x^2 = \frac{4r^2}{5}$.

In the bottom diagram, let the side-length of each square be $y$. Then by Pythagoras' Theorem, $y^2 + y^2 = r^2$ and so $2y^2 = r^2$. So this shaded area is $r^2$.

Therefore the ratio of the two shaded areas is $\frac{4}{5} : 1 = 4 : 5$.

This problem is taken from the UKMT Mathematical Challenges.
View the archive of all weekly problems grouped by curriculum topic

View the previous week's solution
View the current weekly problem