Question

Assertion :$A\equiv (2,2)$, $B\equiv (6,-1)$ and $C\equiv (7,3)$ are the vertices of a $△ABC$. $D,E,F$ are the mid-points of the sides $BC$, $CA$ and $AB$ respectively. Area of the $\Delta DEF$ is equal to $\frac{19}{8}$ square units. Reason: Lines joining the mid-points of the sides of a triangle divide the triangle into four triangles of equal areas.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Reason is true.
Because if $ABC$ is a given triangle and $D,E,F$ are the mid-points of the sides $BC,CA$ and $AB$ respectively, then
Area of $\Delta AEF$.
$=\frac{1}{2}\times \frac{1}{2}BC\times \frac{1}{2}l$ (l being the altitude from A)
$=\frac{1}{4}\times \frac{1}{2}BC\times l=\frac{1}{4}$ area of the triangle $ABC.$
Similarly for triangles $BDF$ and $CDE,$
So that area of $\Delta DEF=\frac{1}{4}$ area of $\Delta ABC$
$=\frac{1}{4}\times \frac{1}{2}\left|\begin{array}{ccc}2& 2& 1\\ 6& -1& 1\\ 7& 3& 1\end{array}\right|=\frac{19}{8}$
and thus assertion is also true.