ABCD is a parallelogram. E is a point on DC extended, such that D and E are on opposite sides of BC. Let AE intersect BC and BD at F and G, respectively. If AG=180 and FG=108, what is EF?
To find the length EF, we need to use the properties of a parallelogram and the given information.
Let's use some properties of parallelograms to help us understand the situation. Since ABCD is a parallelogram, opposite sides are parallel. This means that AE is also parallel to DC.
Now let's use the given information. We know that AG = 180 and FG = 108. Using this information, we can find the length of BF.
Since AE is parallel to DC, we can use similar triangles AEF and CDF. The ratio of corresponding sides of similar triangles is the same.
Let's assume the length of BF is x. By using the properties of similar triangles AEF and CDF, we can write the proportion:
AE / CD = AF / CF
Substituting the lengths, we get:
(AG + GF) / CD = AF / x
Substituting the given values:
(180 + 108) / CD = AF / x
Simplifying:
288 / CD = AF / x
We also know that CD = BC since ABCD is a parallelogram.
Now let's solve for AF:
288 / BC = AF / x
Rearranging the equation:
AF = (288 * x) / BC
Now, let's use the fact that AF + FG = AG:
(288 * x) / BC + 108 = 180
Now we can solve for x:
(288 * x) / BC = 180 - 108
(288 * x) / BC = 72
288 * x = 72 * BC
x = (72 * BC) / 288
Simplifying further:
x = BC / 4
So, we have found that BF = BC / 4.
Now, let's find the length of EF. Since E is a point on DC extended, we have:
EF = BF - BE
Since BF = BC / 4 and BE = BC, we can substitute these values:
EF = (BC / 4) - BC
Combining the fractions:
EF = (BC - 4BC) / 4
EF = -3BC / 4
Therefore, the length of EF is -3BC / 4.