Question

In the figure given, $$D,E$$ are mid-points of the sides $$AB,\ BC$$ respectively and $$DF \parallel BC$$. Prove that $$DBEF$$ is a parallelogram. Calculate $$AC$$ if $$AF=2.6\ cm$$.

Answer 1

Given that,

$$D$$ and $$E$$ are the midpoints of $$AB$$ and $$BC$$ respectively

$$DF || BC$$ and

$$AF =2.6\ cm$$

To prove: $$DBEF$$ is a parallelogram

To find: $$AC$$

Proof:

In $$\triangle ABC$$

$$D$$ is the mid point of $$AB$$ and $$DF \parallel BC$$

$$\Rightarrow F$$ is the mid point of $$AC$$ $$ ...(1)$$ [ Converse of mid point theorem]

Now, $$F$$ and $$E$$ are the mid point of $$AC$$ and $$BC$$

$$\Rightarrow EF \parallel AB$$ [ Mid point theorem]

$$\Rightarrow EF \parallel DB\qquad...(2)$$

Given,

$$DF \parallel BC\Rightarrow DF \parallel BE ....(3)$$

From equation $$(2)$$ and $$(3)$$,

$$DBEF$$ is a parallelogram.

Since, $$F$$ is the midpoint of $$AC$$

$$\therefore AC=2\times AF =2\times 2.6=5.2\ cm$$