Question

Consider an equilateral triangle ABC whose side is 2 cm long.

Let D, E, and F be the midpoints of AB, BC, and AC respectively. Find the area of the rhombus DECF.
A. √3/2 cm2
B. 2 cm2
C. √3 cm2
D. √2 cm2
E. 1/√3 cm2

Answer 1

A

Explanation:

We know that to find the area of the rhombus we need the two diagonals. That means that we need the length of
`DC` and
`EF`.

We can evaluate
`EC` will be 1cm

`FC` will be 1cm

Since
`D` is the midpoint of A to F and
`E` is the midpoint of
`FC` (equilateral),
`DE` is also 1cm.

As
`DF` is parallel to
`EC`, we also know that is 1cm.

As
`E` is also the midpoint of
`DB` and
`F` is the midpoint of
`AD`,
`EF` must also be 1.

Since we have one diagonal we can now work out the other using pythagoras' theorem

Let
`J` be the midpoint of
`EF`

`J` = √(1^2 - 0.5^2)

= √(3/4) = √(3)/2

Therefore
`DJ` is √(3)/2

And
`DC` must then be √3

By using
`DC` and
`EF` we can find the area by p*q/2

(√3* 1)/2

Which is A.