Question

State the whether given statement is true or false

ΔABC, D, E and F are midpoints of sides AB, BC and CA respectively.Prove

$ar\left(BDEF\right)=\frac{1}{2}ar\left(△ABC\right)$

• A. True
• B. False

Answer 1

The correct option is A True

Given :ABC is a Triangle in which the midpoints of sides BC, CA and AB are D, E and F.

To show,$ar\left(BDEF\right)=1/2\left(ABC\right)$

Proof:

We can see that triangle ABC and AFE are similar. Therefore,

$ar\left(AFE\right)/ar\left(ABC\right)=\left(AF{)}^2/\right(AB{)}^2=(1/2{)}^2=1/4$

$ar\left(AFE\right)=ar\left(ABC\right)/4$

Similarly,

$ar\left(CDE\right)=ar\left(ABC\right)/4$

$ar\left(BDF\right)=ar\left(ABC\right)/4$ and

$ar\left(FDE\right)=ar\left(ABC\right)-ar\left(AFE\right)-ar\left(BDF\right)-ar\left(CDE\right)$

$ar\left(FDE\right)=ar\left(ABC\right)-ar\left(ABC\right)/4-ar\left(ABC\right)/4-ar\left(ABC\right)/4$

$ar\left(FDE\right)=ar\left(ABC\right)/4$

Thus, $ar\left(FDE\right)=ar\left(AFE\right)=ar\left(BDF\right)=ar\left(CDE\right)=ar\left(ABC\right)/4$

Now, according to question,

$ar\left(ABC\right)=ar\left(FDE\right)+ar\left(AFE\right)+ar\left(BDF\right)+ar\left(CDE\right)$

$ar\left(ABC\right)=ar\left(BDEF\right)+ar\left(ABC\right)/4+ar\left(ABC\right)/4\left(becausear\right(BDEF)=ar(BDF)+ar(DEF\left)\right)$

$ar\left(BDEF\right)=ar\left(ABC\right)-ar\left(ABC\right)/2$

$ar\left(BDEF\right)=ar\left(ABC\right)/2$