Library / /237 of 297

, 2012

Geometry Difficulty 2.5 Junior Find the answer Canada

The diagram shows a square quilt that is made up of identical squares and two sizes of right-angled isosceles triangles. What percentage of the quilt is shaded?

Hide/Reveal Description of Quilt

A 5 by 5 grid of squares. Each square in the grid is of one of the following four types:

A: unshaded square
B: shaded square
C: square cut in half by a diagonal, with one half shaded
D: square cut into quarters by two diagonals, with two opposite quarters shaded

The types of squares in each of the five rows in the grid is as follows:

A, C, B, C, A
c, A, D, A, C
B, D, A, D, B
C, A, D, A, C
A, C, D, C, A

Pick one

Solution

The quilt consists of 25 identical squares.

Of the 25 squares, 4 are entirely shaded, 8 are shaded with a single triangle that covers half of the square, and 4 are shaded with two triangles that each cover a quarter of the square.

Therefore, the shading is equivalent to the area of $4 + 8 \times \frac{1}{2} + 4 \times 2 \times \frac{1}{4} = 10$ squares.

As a percentage, the shading is $\frac{10}{25}\times 100\% = 40\%$ of the total area of the quilt.

Want a route through all this instead of an archive? The track puts 2,444 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.