O-Level Geometry using coordinates: worked solution

Question

Points A(20, 10) and B(14, 0) lie in the plane. (i) Find the equation of line AB. (ii) If W(9, p) lies on AB, find p.

(i) Equation of line AB Step 1 - find the gradient using m = (y₂ - y₁)/(x₂ - x₁). Take A(20, 10) as the second point and B(14, 0) as the first: m = (10 - 0)/(20 - 14) m = 10/6 Cancel the common factor 2: m = 5/3 Step 2 - substitute a point into y - y₁ = m(x - x₁). B(14, 0) is the easier point because its y-coordinate is 0: y - 0 = (5/3)(x - 14) Step 3 - expand and rearrange into the form y = mx + c. y = (5/3)x - (5/3)(14) (5/3)(14) = 70/3 y = (5/3)x - 70/3 (Check with A: (5/3)(20) - 70/3 = 100/3 - 70/3 = 30/3 = 10, which matches A(20, 10).) (ii) Value of p Step 1 - W(9, p) lies on the line, so its coordinates must satisfy the equation. Substitute x = 9 and y = p. p = (5/3)(9) - 70/3 Step 2 - work out each term. (5/3)(9) = 45/3 = 15 So p = 15 - 70/3 Step 3 - write 15 with denominator 3 so the subtraction can be done. 15 = 45/3 p = 45/3 - 70/3 p = -25/3 So p = -25/3 (that is -8⅓).