Question

In a quadrilateral ABCD with sides a,b,c,d the range of $\frac{a^2+b^2+c^2}{d^2}$ is

• A. $(\frac{1}{3},\infty )$
• B. $(\frac{2}{3},\infty )$
• C. $(1,\infty )$
• D. $(0,\infty )$

Answer 1

Answer: (A)

$(\frac{1}{3},\infty )$

$\text{Solve: Given, ABCD is a Quadrilateral}$

$\text{having sides}a,b,c\text{and}d\text{.}$

$\text{we have to find the Range of}$

$\frac{a^2+b^2+c^2}{d^2}$

since $\sum (a-b{)}^2⩾0$

$\Rightarrow (a-b{)}^2+(b-c{)}^2+(c-a{)}^2⩾0$

$\Rightarrow 2(a^2+b^2+c^2)⩾2(ab+bc+ca)$

on adding $a^2+b^2+c^2$ to both sides we get

$3(a^2+b^2+c^2)⩾(a+b+c{)}^2>d^2$

$\Rightarrow \frac{a^2+b^2+c^2}{d^2}>\frac{1}{3}$

Range $(\frac{1}{3},\infty )$