# Problem of the Week Problem C and Solution Tile Art

## Problem

A tile measuring $$8$$ cm by $$8$$ cm has gridlines drawn on it, parallel to each side and spaced $$1$$ cm apart. Six blue triangles are then painted on the tile, as shown.

What fraction of the tile is painted blue?

## Solution

We will start by determining the areas of the six painted triangles. We label the triangles $$A$$, $$B$$, $$C$$, $$D$$, $$E$$, and $$F$$ and draw in a height and a base for each triangle.

We will calculate the area of each triangle using the formula for the area of a triangle: $\text{area} = \frac{\text{base} \times \text{height}}{2}$

Triangle $$A$$ has base $$2$$ cm and height $$3$$ cm. The area of triangle $$A$$ is then $$\frac{2\times 3}{2}=\frac{6}{2}=3 \text{ cm}^2$$.

Triangle $$B$$ has base $$3$$ cm and height $$4$$ cm. The area of triangle $$B$$ is then $$\frac{3\times 4}{2}=\frac{12}{2}=6 \text{ cm}^2$$.

Triangle $$C$$ has base $$3$$ cm and height $$4$$ cm. The area of triangle $$C$$ is then $$\frac{3\times 4}{2}=\frac{12}{2}=6 \text{ cm}^2$$.

Triangle $$D$$ has base $$2$$ cm and height $$3$$ cm. The area of triangle $$D$$ is then $$\frac{2\times 3}{2}=\frac{6}{2}=3 \text{ cm}^2$$.

Triangle $$E$$ has base $$4$$ cm and height $$2$$ cm. The area of triangle $$E$$ is then $$\frac{4\times 2}{2}=\frac{8}{2}=4 \text{ cm}^2$$.

Triangle $$F$$ has base $$2$$ cm and height $$4$$ cm. The area of triangle $$F$$ is then $$\frac{2\times 4}{2}=\frac{8}{2}=4 \text{ cm}^2$$.

The total area painted blue is then $$3+6+6+3+4+4=26 \text{ cm}^2$$.

The area of the entire tile is $$8 \times 8 = 64 \text{ cm}^2$$.

Thus, $$\frac{26}{64}=\frac{13}{32}$$ of the tile is painted blue.