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Enclosing Squares

Can you find sets of sloping lines that enclose a square?

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Parallel Lines

How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?

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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?

Graphical Triangle

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

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. This gives $300 - (112½ + 12½ + 100) = 75$.

This problem is taken from the UKMT Mathematical Challenges.