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Perpendicular Lines

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

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Surprising Transformations

I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?

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Painting Between the Lines

In abstract and computer generated art, a real object can be represented by a simplified set of lines. Can you create a picture using mathematical instructions?

Graphical Triangle

Age 14 to 16 Short 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 $(5,1)$.

Subtracting Triangles From A Rectangle:
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 $300 - (112½ + 12½ + 100) = 75$.

Subtracting A Triangle From A Larger Triangle:
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.