Question

Assertion :${AA}_1,{BB}_1,{CC}_1$ are the medians of triangle $ABC$ whose centroid is $G$.If the points $A,C_1,G$ and $B$ are concyclic, then $c^2,a^2,b^2$ are in A.P. Reason: $BG.{CC}_1={BC}_1.BA$

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

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

$\because A,C_1,G$ and $B_1$ are concyclic(given), then
$BG.{BB}_1={BC}_1.BA$
$\Rightarrow \frac{2}{3}{BB}_1.{BB}_1=\frac{c}{2}.c$
$\Rightarrow \frac{2}{3}{\left({BB}_1\right)}^2=\frac{c^2}{2}$
$\frac{2}{3}\left(\frac{2a^2+2c^2-b^2}{4}\right)=\frac{c^2}{2}$
$\Rightarrow 2a^2+2c^2-b^2=3c^2$
or $2a^2=b^2+c^2$
$\therefore c^2,a^2,b^2$ are in A.P.