Challenge Level

The equations of the three lines must be considered in pairs to find the coordinates of their points of intersection, i.e. the coordinates of the vertices of the triangle. By solving each pair of simultaneous equations, you should find that the coordinates are $(-15,-9)$, $(0,6)$ and $(5,1)$.

The area of the triangle may now be found by enclosing the triangle in a rectangle and subtracting the areas of the three surrounding triangles from the area of the rectangle, i.e Yellow Triangle = Rectangle - (Red Triangle + Blue Triangle + Green Triangle). This gives $300 - (112 ½ + 12 ½ + 100) = 75$.

The area of the triangle may be found by caluculating the area of the large triangle and subtracting the area of the 'extra' triangle, i.e. Red Triangle = Combined Triangle - Blue Triangle. This gives $225-150=75$.

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.