IMC 2005 Q23
23. What is the area (in square units) of the triangle formed by the three lines whose equations are: \(y - x = 6\), \(x - 2y = 3\), \(x + y = 6\) ?
- A 55
- B 60
- C 65
- D 70
- E 75
Intermediate Mathematical Challenge 2005, Question 23. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 75
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. It is left to the reader to show that these are (−15, −9), (0, 6) and (5, 1). The area of the triangle may now be found by enclosing it in a rectangle measuring 20 × 15 and subtracting the areas of the three surrounding triangles from that of the rectangle. This gives 300 − (112\(\tfrac{1}{2}\) + 12\(\tfrac{1}{2}\) + 100) = 75.
Solution © UK Mathematics Trust, from the IMC 2005 solutions.