Tag: construction of parallel lines and triangles

Questions Related to construction of parallel lines and triangles

O is a point that lies in the interior of $\Delta ABC$. Then $2(OA - OB -OC) > \text{Perimeter}\ of\ \Delta ABC$.

maths construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle triangle inequality

  1. True

  2. False

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Correct Option: B
Explanation:
From the $\triangle ABC,$ by triangle inequality,
$ OA+OB>AB$ ....... $(i)$
$ OB+OC>BC$ ........ $(ii)$
$ OA+OC>AC$ ........ $(iii)$
By adding $(i),(ii)$ and $(iii)$
$ 2(OA+OB+OC)>AB+BC+AC$
$ \therefore 2(OA+OB+OC)>\text{Perimeter of triangle } ABC$
Hence, the statement is false.

Sum of the length of any two sides of a triangle is always greater than the length of third side.

maths construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle triangle inequality

  1. True

  2. False

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Correct Option: A
$\dfrac{\sin 2x}{2\cos x}=\tan x \ \ ?$
maths construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle triangle inequality

  1. True

  2. False

Show answer
Correct Option: A
Explanation:
LHS

$\dfrac{\sin 2x}{2\cos x}$

$\dfrac{2\sin x \cos x}{2\cos x}$

$\implies \dfrac{\sin x}{\cos x}$

$\implies \tan x $

Hence proved.

The points $\left( 0,\dfrac { 8 }{ 3 }  \right),(1,3)$ and $(82,30)$ are the vertices of:

maths construction of parallel lines and triangles triangle inequality inequalities in triangle inequalities in triangles

  1. an equilateral triangle

  2. an isosceles triangle

  3. a right angled triangle

  4. none of these

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Correct Option: A
Explanation:

According to the problem :

$AB^2=(0-1)^2+(\dfrac{8}{3}-3)^2$
$=1+\dfrac{1}{9}=\dfrac{10}{9}=1.11$

Similarly,
$BC^2=(82-1)^2+(30-3)^2=7290$
and
$AC^2=(82-0)^2+(30-\dfrac{8}{3})^2=7471.11$

Therefore,
$AB^2+BC^2<AC^2$

Hence the answer is acute-angled triangle.

In $\Box PQRS$,  $PQ+QR+RS+SP> PR+QS$.

maths construction of parallel lines and triangles triangle inequality inequalities in triangle inequalities in triangles

  1. True

  2. False

Show answer
Correct Option: A