Question

State true or false:

$D,E$ and $F$ are the mid-points of the sides $AB,BC$ and $CA$ of an isosceles $△ABC$ in which $AB=BC$. then
$△DEF$ is also isosceles.

• A. True
• B. False

Answer 1

The correct option is A True

AB = AC
Hence, $\angle ABC=\angle ACB$ (Isosceles triangle property)
Now, since, D and F are mid point of AB and AC respectively, thus DF II BC (Mid point theorem)
Hence,
$\angle ADF=\angle ABC$ and $\angle AFD=\angle ACB$ (Corresponding angles)
Thus, $\angle ADF=\angle ABC=\angle AFD=\angle ACB$
Now, In $△$ ADF and FEC
$\angle ADF=\angle FEC$ (Corresponding angles of parallel lines EF and AB)
$\angle AFD=\angle ACB$(Corresponding angles of parallel lines DF and BC)
AF = FC (F is the mid point of AC)
Thus $△ADF\cong △FEC$ (AAS rule)
Hence, AD = FE (corresponding sides of congruent triangles)
Similarly, we can prove, AF = DE
Since, AD = AF (half lengths of equal sides, AB and AC)
Thus, EF = DE or $△$ DEF is an isosceles triangle.