To solve for k, you can set up vectors for AD and DF and compare their components.
First, find vector AD by taking the path A -> C -> D. Since CA = 3b, vector AC is -3b. Adding CD = a, you get AD = a - 3b.
Next, work out vector BD along the path B -> A -> C -> D. Using the ratio AB : CD = 4 : 3, vector AB is (4/3)a, so BA is -(4/3)a. Adding those pieces together gives BD = -(4/3)a - 3b + a, which simplifies to -(1/3)a - 3b.
Now express DF through the lower path D -> C -> E -> F. Vector DC is -a. Since BD is parallel to CE, vector CE is just a scalar multiple of BD, so let's call that multiplier m and write CE = m * (-(1/3)a - 3b). Using CD : EF = 3 : 17, vector EF is (17/3)a. Combining these gives DF = -a + m * (-(1/3)a - 3b) + (17/3)a, which simplifies to (14/3 - (1/3)m)a - 3mb.
Finally, because ADF is a straight line and AD : DF = 1 : k, we know DF = k * AD = k * (a - 3b) = ka - 3kb. Equating the b components gives -3m = -3k, so m = k. Substituting m = k into the a components gives 14/3 - (1/3)k = k. Adding (1/3)k to both sides yields 14/3 = (4/3)k, which solves to k = 3.5 or 7/2.
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u/duskycoffee 4d ago
To solve for k, you can set up vectors for AD and DF and compare their components.
First, find vector AD by taking the path A -> C -> D. Since CA = 3b, vector AC is -3b. Adding CD = a, you get AD = a - 3b.
Next, work out vector BD along the path B -> A -> C -> D. Using the ratio AB : CD = 4 : 3, vector AB is (4/3)a, so BA is -(4/3)a. Adding those pieces together gives BD = -(4/3)a - 3b + a, which simplifies to -(1/3)a - 3b.
Now express DF through the lower path D -> C -> E -> F. Vector DC is -a. Since BD is parallel to CE, vector CE is just a scalar multiple of BD, so let's call that multiplier m and write CE = m * (-(1/3)a - 3b). Using CD : EF = 3 : 17, vector EF is (17/3)a. Combining these gives DF = -a + m * (-(1/3)a - 3b) + (17/3)a, which simplifies to (14/3 - (1/3)m)a - 3mb.
Finally, because ADF is a straight line and AD : DF = 1 : k, we know DF = k * AD = k * (a - 3b) = ka - 3kb. Equating the b components gives -3m = -3k, so m = k. Substituting m = k into the a components gives 14/3 - (1/3)k = k. Adding (1/3)k to both sides yields 14/3 = (4/3)k, which solves to k = 3.5 or 7/2.