Tag: triangle inequality

Questions Related to triangle inequality

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

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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.

If $O$ is any point in the interior of $\Delta ABC$. then "$2(OA+OB+OC)=(AB+BC+CA)$" the statement  is?

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. True

  2. False

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Correct Option: A

In $\triangle {ABC},ABC,APQ$ and $\overline { PQ } \parallel \overline { BC } $. If $PQ=5,AP=4,AB=12$, then $BC=$_____

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. $9.6$

  2. $20$

  3. $15$

  4. $5$

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Correct Option: C

If a,b,c are the sides of a triangle ABC, then $\sqrt{a} + \sqrt{b} - \sqrt{c} $  is always:

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. negative

  2. Positive

  3. non - negative

  4. non - positive

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Correct Option: B
Explanation:

If $a,b$ and $c$ are the sides of the triangle then $\sqrt a  + \sqrt b  - \sqrt c $ it always positive because the sum of two sides of the triangle is always greater than the third side.

 

In a $\Delta ABC$, side AB has the equation $2x+3y=29$ and the side AC has the equation, $x+2y=6$. If the mid-point of BC is (5, 6), then the equation of BC is

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. $x-y=-1$

  2. $5x-2y=13$

  3. $21x+31y=291$

  4. $3x-4y=-9$

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Correct Option: A

In a triangle ABC , if AB , BC and AC are the three sides of the triangle , then which of the following statements is necessarily true ?

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. $\displaystyle AB + BC < AC$

  2. $\displaystyle AB + BC > AC$

  3. $\displaystyle AB + BC = AC$

  4. $\displaystyle AB^2 + BC^2 = AC^2$

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Correct Option: B
Explanation:
The sum of any two sides of a triangle is greater than the third side .

In $\triangle ABC, AB, BC$ and $AC$ are the three sides ,

Now ,

$AB + BC > AC$

For a triangle ABC, which of the following is true?

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. ${ BC }^{ 2 }-{ AB }^{ 2 }={ AC }^{ 2 }$

  2. $AB-\,AC=BC$

  3. $(AB-\,AC)>BC$

  4. $ AB - AC< BC$

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Correct Option: D
Explanation:

For any $\triangle ABC$,
the difference between two sides of the triangle should be less than the third side.
Thus, $AB - AC < BC$

In $\triangle PQR$, if $\angle R\displaystyle>\angle Q$, then

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. $QR>PR$

  2. $PQ>PR$

  3. $PQ

  4. $QR

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Correct Option: B
Explanation:

In $\triangle PQR$, $\angle R > \angle Q$
Hence, $PQ > PR$ (Sides opposite larger angles is large)

option B is correct