Question

The line segments joining the midpoints of the sides of a triangle form four triangles each of which is:

• A. similar to the original triangle
• B. congruent to the original triangle
• C. an equilateral triangle
• D. an isosceles triangle

Answer 1

The correct option is A similar to the original triangle

Given: $△ABC$, D, E and F are mid points of AB, BC, CA respectively.
In $△ABC$
F is mid point of AC and D is mid point of AB.
Thus, By Mid point theorem
$FD=\frac{1}{2}CB$ and $FD=CB$ or $FD=CE$ and $FD\parallel CE$ (1)

Similarly,
$DE=FC$ and $DE\parallel FC$ (2)
$FE=DB$ and $FE\parallel DB$ (3)
From (1), (2) and (3)
$\square ADEF$, $\square DBEF$ and $\square DECF$ are parallelograms
The diagonal of a parallelogram divides the parallelogram into two congruent triangles.
Hence, $△DEF\cong △ADF$
$△DEF\cong △DBE$
$△DEF\cong △FEC$
or, $△DEF\cong △ADF\cong △ECF\cong △ADF$
thus, mid points divide the triangle into 4 equal parts.
Thus, the smaller triangles are congruent to each other and similar to the original triangle.