SOLUTION: D and E ARE POINTS ON THE SIDES AB AND AC RESPECTIVELY OF TRIANGLE ABC SUCH THAT DE IS PARALLEL TO BC, AND AD:DB = 4:5. CD AND BE INTERSECT EACH OTHER AT F. FIND THE RATIO OF THE A

Algebra ->  Triangles -> SOLUTION: D and E ARE POINTS ON THE SIDES AB AND AC RESPECTIVELY OF TRIANGLE ABC SUCH THAT DE IS PARALLEL TO BC, AND AD:DB = 4:5. CD AND BE INTERSECT EACH OTHER AT F. FIND THE RATIO OF THE A      Log On


   

Question 981361: D and E ARE POINTS ON THE SIDES AB AND AC RESPECTIVELY OF TRIANGLE ABC SUCH THAT DE IS PARALLEL TO BC, AND AD:DB = 4:5. CD AND BE INTERSECT EACH OTHER AT F. FIND THE RATIO OF THE AREAS OF TRIANGLE DEF AND TRIANGLE BCF .
Found 2 solutions by mananth, KMST:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!

DE || BC
AD/DB = AE/EF =4/5 ( Basic proportionality theorem)
In triangle ABE & ACD

AD/DB = AE/EC and angle A is common
So triangles ABE & ACD are similar
Therefore AD/DB = EF/FB ( properties of similar triangles)
Similarly AE / EC = BF/FC
In Triangles AEF & BFC
angle FEC is congruent to angle BFE ( alternate angles)
angel DEF is = angle BFC ( vertically opposite angles)
Therefore they are similar
Area of triangle DFE / area of BFC = (4/5)^2 ( Properties of similar triangles
= 16/25


Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!


We can prove that
triangles ADE and ABC are similar and their corresponding sides' lengths are in the ratio 4:9,
and that
triangles DEF and CBF are similar and their corresponding sides' lengths are in the ratio 4:9.
Since for similar figures the ratio of the areas is the square of the ratio of their corresponding sides' lengths,
the ratio of areas of DEF and CBF is %284%2F9%29%5E2=highlight%2816%2F81%29 .
The ratio of areas of triangle DEF and triangle BCF is highlight%2816%3A81%29 .

PROOF DETAILS:
AD%2FDB=4%2F5=0.8-->AD=0.8DB-->
Angles BAC and DAE are congruent because they are the same angle.

Angles ABC and ADE are congruent because they are corresponding angles formed by parallel lines BC and DE with transversal AB.
Angles ACB and AED are congruent because they are corresponding angles formed by parallel lines BC and DE with transversal AC.
Triangles ABC and ADE are similar, because they have three pairs of congruent angles.
Since the triangles are similar the ratios of corresponding sides are the same,
and since AD%2FAB=4%2F9 , then DE%2FBC=4%2F9 .
Angle DEF is congruent with angle CBF because they are alternate interior angles formed by parallel lines BC and DE with transversal EB.
Angle EDF is congruent with angle BCF because they are alternate interior angles formed by parallel lines BC and DE with transversal EB.
Angle DFE is congruent with angle CF because they are vertical angles.
Triangles DEF and CBF are similar, because they have three pairs of congruent angles.
Since the triangles are similar the ratios of corresponding sides are the same:
DE%2FBC=4%2F9 .
Since for similar figures the ratio of the areas is the square of the ratio of their corresponding sides' lengths,
the ratio of areas of DEF and CBF is %284%2F9%29%5E2=highlight%2816%2F81%29 .
The ratio of areas of triangle DEF and triangle BCF is highlight%2816%3A81%29 .