SATHEE: Properties of Triangles 2 Question 9

Properties of Triangles 2 Question 9

9. Show that for any triangle with sides $a, b, c$ $3 (a b + b c + c a) \leq (a + b + c)^{2} \leq 4 (a b + b c + c a)$ .

$(1979, 3 M)$

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Solution:

By using triangular inequality,

$c < a + b$

$c^{2} < c a + a b$

Similarly, $a^{2} < a b + a c$ and $b^{2} < b c + a b$

$\begin{array}{lc} ∴ & a^{2} + b^{2} + c^{2} < 2 a b + 2 b c + 2 c a \\ \Rightarrow & (a^{2} + b^{2} + c^{2}) + 2 a b + 2 b c + 2 c a < 4 (a b + b c + c a) \\ \Rightarrow & (a + b + c)^{2} < 4 (a b + b c + c a) \end{array}$

Now, using AM-GM inequality in $a, b$ and $c$ , we get

$\begin{aligned} \frac{a^{2} + b^{2}}{2} \geq a b, \frac{b^{2} + c^{2}}{2} \geq b c and \frac{c^{2} + a^{2}}{2} \geq c a \\ \Rightarrow & a^{2} + b^{2} + c^{2} \geq a b + b c + c a \\ \Rightarrow & a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a \geq 3 (a b + b c + c a) \\ \Rightarrow & (a + b + c)^{2} \geq 3 (a b + b c + c a) \end{aligned}$

From Eqs. (i) and (ii), we get

$3 (a b + b c + c a) \leq (a + b + c)^{2} < 4 (a b + b c + c a)$

Properties of Triangles 2 Question 9

9. Show that for any triangle with sides $a, b, c$ $3 (a b + b c + c a) \leq (a + b + c)^{2} \leq 4 (a b + b c + c a)$ .

Solution:

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Properties of Triangles 2 Question 9

9. Show that for any triangle with sides a,b,c 3(ab+bc+ca)≤(a+b+c)2≤4(ab+bc+ca).

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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9. Show that for any triangle with sides $a, b, c$ $3 (a b + b c + c a) \leq (a + b + c)^{2} \leq 4 (a b + b c + c a)$ .