Graphical triangle
What is the area of the triangle formed by these three lines?
Age
14 to 16
Challenge level
Problem
What is the area of the triangle formed by the three lines whose equations are given below?$$y-x=6\qquad x-2y=3\qquad x+y=6$$
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
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 $(5,1)$.
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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 $300 - (112 ½ + 12 ½ + 100) = 75$.
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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 $225-150=75$.