Question

If $\Delta ABC$ is an isosceles triangle and midpoints $D,E,$ and $F$ of $AB,BC,$ and $CA$ respectively are joined, then $\Delta DEF$ is:

• A. Equilateral
• B. Isosceles
• C. Scalene
• D. Right-angled

Answer 1

Answer: (B)

Isosceles

Given: In $△ABC,D,E$ and $F$ are midpoints of sides $AB,BC$ and $CA$.

$BE=EC$

$\therefore DF=\frac{1}{2}BC$

$\therefore \frac{DF}{BC}=\frac{1}{2}$ ....... $\left(1\right)$

Similarly, $\frac{DE}{AC}=\frac{1}{2}$ and $\frac{EF}{AB}=\frac{1}{2}$

$\Rightarrow \frac{DF}{BC}=\frac{DE}{AC}=\frac{EF}{AB}=\frac{1}{2}$

$△ABC$ is propotional to $△DEF$ as the sides of the triangles are proportional.

So corresponding angles are equal.

Hence, $△ABC\sim △EDF$ [by SSS similarly theorm]

$\therefore △DEF$ is isosceles.