Solving Geometric Problems with Straight Lines
Explore a unique geometric problem involving straight lines and triangles. Follow the step-by-step process to find the areas of red and blue triangles, determine the numerical area of the intersection, and derive the equation of the line using coordinate geometry. Illustrative images aid in understanding the problem.
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Straight Line From a suggestion by Mark Richards (Lancaster Girls Grammar School) who got it from Bernard Murphy (MEI)
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 7 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 6 (not to scale)
Straight Line ? 1 2?? area of the red triangle: 1 2?? area of the blue triangle: ? 1 2?? area of the red and blue triangle: ? ?,? ? 1 2?? +1 2?? =1 equating areas: 2?? ?? + ?? = ?? ? = ? equation of the line: ? ?? + ? ?? = ?? + ?? ?? + ?? = ?? ? ? (not to scale)
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 7 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 6 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 7 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 5 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 8 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 6 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 6 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 8 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 3 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 5 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 5 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 8 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 2 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 5 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 11 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 8 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 6 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 5 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 7 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 10 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 12 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 5 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 12 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 6 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 12 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 11 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 12 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 13 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 13 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 4 (not to scale) SIC_31
Straight Line ? The point ? has coordinates (?,?) Find, in terms of ?, the area of the red triangle. 17 Find, in terms of ?, the area of the blue triangle. Find the numerical area of the triangle which is red and blue. ? ?,? Hence find an equation linking ? and ?. What is the equation of the line? ? 4 (not to scale) SIC_31
