In ABC, if a, b, c are in A.P. (with usual notations), identify t… - Vidyadip

MCQ

In $\Delta ABC$, if $a, b, c$ are in $A.P.$ (with usual notations), identify the incorrect statements -

A
$h_1, h_2, h_3$ are in $H.P.$, where $h_1, h_2, h_3$ are altitudes from vertices $A,B$ $C$ respectively.
B
$sinA, sinB, sinC$ are in $A.P.$
✓
$r_1, r_2, r_3$ are in $A.P.$
D
$tan \frac{A}{2} , tan \frac{B}{2}, tan \frac{C}{2} $ are in $H.P.$

✓

Answer

Correct option: C.

$r_1, r_2, r_3$ are in $A.P.$

c
$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}.$

$\lambda \mathrm{a}, \lambda \mathrm{b}, \lambda \mathrm{c}$ will be in $\mathrm{AP}.$

$\Rightarrow \sin A, \sin B, \sin C$ in $A P.$

Also $\mathrm{h}_{1}=\frac{2 \Delta}{\mathrm{a}} ; \mathrm{h}_{2}=\frac{2 \Delta}{\mathrm{b}} ; \mathrm{h}_{3}=\frac{2 \Delta}{\mathrm{c}}$ will be in $HP.$

$a, b, c$ in $A P \Rightarrow s-a, s-b, s-c$ in $A P$

$\Rightarrow \frac{\Delta}{s-a}, \frac{\Delta}{s-b} ; \frac{\Delta}{s-c}$ will be in $HP.$

$\Rightarrow \mathrm{r}_{1}, \mathrm{r}_{2}, \mathrm{r}_{3}$ in $H.P.$

Also $\frac{\Delta}{s(s-a)}, \frac{\Delta}{s(s-b)}, \frac{\Delta}{s(s-c)}$ will be in $HP.$

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