Question

In parallelogram ABCD with CD=a and CA=b, E is the point on CD such that CE:ED=1:3 and F is the point on CA such that CF:FA=1:4. Work out, in its simplest form, the ratio BE:BF. Show all your working.

Answer 1

To find the ratio BE:BF in the parallelogram, we can divide CD and CA into smaller parts. By doing so, we can find the lengths of CE, ED, CF, and FA. BE:BF = 15a : 16b

Step-by-step explanation:

First, we can find the lengths of CE and ED by dividing CD (a) into four equal parts. So, CE = a/4 and ED = 3a/4. Similarly, we can find the lengths of CF and FA by dividing CA (b) into five equal parts. Thus, CF = b/5 and FA = 4b/5.

Next, we can find BE by subtracting CE from CD. BE = CD - CE = a - a/4

= 3a/4.

Similarly, we can find BF by subtracting CF from CA. BF = CA - CF

= b - b/5

= 4b/5.

Therefore, the ratio BE:BF is 3a/4 : 4b/5. To simplify this ratio, we can multiply both sides by the least common multiple of the denominators (4 and 5), which is 20. Multiplying and simplifying, we get:

BE:BF = 15a : 16b