practice problems pending

 Q1. ABC is a triangle and D and E are interior points of the sides AB and BC respectively such that:

AD/DB = 1/3
CE/EB = 3

If AE and CD intersect at F, find CF/FD.

S1.



Approach 1: Using mass points:
AD/DB = 1/3
=> A has higher mass since it's closer to D.
Let mA = 3 mB = 1
=> mC = 1/3
=> mE = 4/3

Now mF = 3 + 4/3 = 13/3 and it matches 4 + 1/3 = 13/3
So we have assigned masses correctly.

=> CF/FD = 4/(1/3) = 12

Approach 2: Menelaus theorem
Typically you would notice that if we have a solution using mass points, we can also solve it using Ceva's theorem or Menelaus' theorem.

Here, in triangle BDC, AFE is the traversal intersecting all sides.
BE/EC * CF/FD * DA/AB = 1

=> 1/3 * CF/FD * 1/4 = 1

=> CF/FD = 12 = Answer

Q2. L and M are the mid-points of the diagonals BD and AC respectively of the quadrilateral ABCD. Through D, draw DE equal and parallel to AB. Show that

EC || LM
EC = 2LM


S2.
It's straightforward with co-ordinate geometry.
A (0,0)
B(x1,0)
C(x3,y3)
D(x2,y2)


Since DE || BC and DE = BC
=> E = (x2 + x1,y2)

Now compute L,M co-ordinates and slope of LM.
You will see that slope is same as EC.

Also compute their lengths, you will see that EC = 2LM.
H.P.



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