Let ABC be triangle in the xy plane, where B is at the origin (0, 0). Let BC be produced to D such that BC : CD = 1 : 1, CA be produced to E such that CA : AE = 1 : 2 and AB be produced to F such that AB : BF = 1 : 3. Let G(32, 24) be the centroid of the triangle ABC and K be the centroid of the triangle DEF. Find the length GK.
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. BC : CD = 1 : 1 hence D = (2x1, 2y1).
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. CA : AE = 1 : 2 hence E = (3x2 – 2x1, 3y2 – 2y1).
Hint 3: Proceed with the final algebraic steps to solve the system. AB : BF solve for the final value : 3 hence F = (–3x2, –3y2).
Step 1: BC : CD = 1 : 1 hence D = (2x1, 2y1)
Step 2: CA : AE = 1 : 2 hence E = (3x2 – 2x1, 3y2 – 2y1)
Step 3: AB : BF = 1 : 3 hence F = (–3x2, –3y2)
Step 4: => Centroid of DEF = K = (0, 0)
Step 5: Centroid of ABC = G = (32, 24)
Step 6: => GK = 322 + 242 = 40
Ready to track your progress and master these topics?
Create a free account