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Congruence of Triangles and Inequalities in a Triangle — Maths STD 9 — Question

Maharashtra BoardEnglish MediumSTD 9MathsCongruence of Triangles and Inequalities in a Triangle5 Marks
Question
In the given figure, prove that
  1. CD + DA + AB > BC
  2. CD + DA + AB + BC > 2AC.
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✓

Answer


Given: Quadrilateral ABCD
To prove:
  1. CD + DA + AB > BC
  2. CD + DA + AB + BC > 2AC
Proof:
  1. In $\triangle\text{ACD},$
$\text{CD + DA > CA }...(1)$

In $\triangle\text{ABC,}$

$\text{AB + CA > BC }...(2)$

Adding (1) and (2), we get

$\text{CD + DA + AB +CA > CA + BC}$

$\therefore\text{AB + CD + DA + BC}$
  1. In $\triangle\text{CDA,}$
$\text{CD + DA + >CA }...(3)$

In $\triangle\text{BCA},$

$\text{BC + AB + CA }...(4)$

Adding (3) and (4), we get

$\text{CD + AD + BC + AB > CA + CA}$

$\therefore\text{CD + AD + BC + AB > 2CA}$

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