Tag: triangle inequality

Questions Related to triangle inequality

In $\Delta ABC$, if $\angle A = 50^{\circ}$ and $\angle B = 60^{\circ}$, then the greatest side is :

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

  1. AB

  2. BC

  3. AC

  4. Cannot say

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Correct Option: A
Explanation:

Using angle sum property of triangle

$\angle A+\angle B+\angle C={ 180 }^{ \circ  }\ \Rightarrow { 50 }^{ \circ  }+{ 60 }^{ \circ  }+\angle C={ 180 }^{ \circ  }\ \Rightarrow \angle C={ 180 }^{ \circ  }-{ 110 }^{ \circ  }={ 70 }^{ \circ  }$
Side opposite to the largest angle is the longest side.
Here $\angle C$ is largest and side opposite to it is $AB$
So $AB$ is the longest side.

In $\Delta ABC$, if $\angle A = 35^{\circ}$ and $\angle B = 65^{\circ}$, then the longest side of the triangle is :

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

  1. AC

  2. AB

  3. BC

  4. None of these

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Correct Option: B
Explanation:

Using angle sum property of triangle

$\angle A+\angle B+\angle C={ 180 }^{ \circ  }\ \Rightarrow { 35 }^{ \circ  }+{ 65 }^{ \circ  }+\angle C={ 180 }^{ \circ  }\ \Rightarrow \angle C={ 180 }^{ \circ  }-{ 100 }^{ \circ  }={ 80 }^{ \circ  }$
Side opposite to largest angle is the longest side of any triangle.
Here $\angle C$ is the largest angle and side opposite to it is $AB$
So $AB$ is the longest side.

In $\Delta ABC$, if AB $>$ BC then :

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

  1. $\angle C < \angle A$

  2. $\angle C = \angle A$

  3. $\angle C > \angle A$

  4. $\angle A = \angle B$

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Correct Option: C
Explanation:

In any triangle angle opposite to the longest side is largest.

Angle opposite to $AB$ is $\angle C$ and angle opposite to $BC$ is $A$
Here $AB>AC$
$\Rightarrow \angle C>\angle A$
So option $C$ is correct.

If length of the largest side of a triangle is 12 cm then other two sides of triangle can be :

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

  1. 4.8 cm, 8.2 cm

  2. 3.2 cm, 7.8 cm

  3. 6.4 cm, 2.8 cm

  4. 7.6 cm, 3.4 cm

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Correct Option: A
Explanation:

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

Here the sum must be greater than $12\ \ cm$
In option $A$
$4.8\ \ cm+8.2\ \ cm=13\ \ cm$
$\Rightarrow 13\ \ cm>12 \ \ cm$
In rest of the options sum is less than $12\ \ cm$
So option $A$ is correct. 

In $\Delta ABC, \angle A=100^{\circ}, \angle B=30^{\circ}$ and $\angle C= 50^{\circ}$,then

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

  1. $AB>AC$

  2. $AB=AC$

  3. $AB<AC$

  4. None of these

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Correct Option: A
Explanation:
In any triangle side opposite to the largest angle is the longest side.
Here $\angle A$ is largest and side opposite to it is $BC$
$\therefore BC$ is the longest side.
Then comes $\angle C$ and side opposite to it is $AB$
$\therefore AB$ is the second longest side.
Then comes $\angle B$ and side opposite to it is $AC$
$\therefore AC$ is the shortest side.
So the increasing order of sides is
$AC<AB<BC$
$\Rightarrow AB>AC$
So option $A$ is correct. 

Out of isosceles triangles with sides of 7 cm and a base with the length expressed by whole number, the triangle with the greatest perimeter was selected. This perimeter is equal to.......

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

  1. 14 cm

  2. 15 cm

  3. 21 cm

  4. 27 cm

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Correct Option: D
Explanation:
Since sum of the two sides is greater than the third side.

$7+7>x$    [for a triangle]

for max perimeter, $x=13$

$\therefore$   perimeter $=7+7+13=27\ cm$

If a $\triangle PQR$ is constructed taking QR = $5$ cm, PQ = $3$ cm and PR = $4$ cm, then the correct order of the angles of the triangle is:

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

  1. $\displaystyle \angle P$ < $\displaystyle \angle Q$ < $\displaystyle \angle R$

  2. $\displaystyle \angle P$ > $\displaystyle \angle Q$ < $\displaystyle \angle R$

  3. $\displaystyle \angle P$ > $\displaystyle \angle Q$ >$\displaystyle \angle R$

  4. $\displaystyle \angle P$ < $\displaystyle \angle Q$>$\displaystyle \angle R$

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Correct Option: C
Explanation:

In a triangle, the angle is determined by their sides if it is given. The largest side will have the largest angle opposite it. The smallest side will have the smallest angle opposite to it.


So, $QR=5\ cm$. It is the largest side. Hence the angle opposite to it will also be largest that is$\angle P.$


Then the side$PR=4\ cm$, smaller than $QR$. Hence the $\angle Q$ will be smaller than $\angle P$

Finally, the smallest side $PQ=3\ cm$ with its corresponding angle $\angle R$ is smallest.

Hence the option C is right.

If a triangle $PQR$ has been constructed taking $QR = 6 $ cm, $PQ = 3 $ cm and $PR = 4 $ cm, then the correct order of the angle of triangle is

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

  1. $\displaystyle \angle P< \angle Q< \angle R $

  2. $\displaystyle \angle P> \angle Q< \angle R $

  3. $\displaystyle \angle P> \angle Q> \angle R $

  4. $\displaystyle \angle P< \angle Q> \angle R $

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Correct Option: C
Explanation:

Given, in $\triangle PQR$, $QR=6$ cm, $PQ=3$ cm, $PR=4$ cm

We know, 
(i) the shortest side is always opposite the smallest interior angle.

(ii) the longest side is always opposite the largest interior angle.
Here, $QR=6$ cm is the largest side, therefore $\angle P$ is the greatest.

And $PQ=3$ cm is the smallest side, therefore $\angle R $ is the smallest angle.
Therefore, the correct order is $\angle P>\angle Q>\angle R$.

The number of triangles with any three of the length $1, 4, 6$ and $8 $ cm as sides is:

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

  1. $4$

  2. $2$

  3. $1$

  4. $0$

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Correct Option: C
Explanation:

Only $1.$ Since, the sum of any two sides of a triangle must be greater than the third side.

$1,4,6$ no, because $1+4<6$
$1,4,8$ no, because $1+4<8$
$1,6,8$ no, because $1+6<8$
$4,6,8$ yes, because $4+6>8 , 4+8>6 , 8+6>4$
Option $C$ is correct.

Which of the following sets of side lengths will not form a triangle?

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

  1. $11$ cm, $10$ cm, $11$ cm

  2. $3$ m, $3$ m , $3$ m

  3. $9$ mm, $9$ mm, $12$ mm

  4. $3$ cm, $4$ cm, $7$ cm

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Correct Option: D
Explanation:

The sum of any two sides of a triangle is greater than the third side. 

Here, if we consider $3$ cm, $4$ cm, $7$ cm as side lengths then the sum of two sides $(3 + 4)$ cm is equal to the third side and not greater than the third side i.e., $7$ cm.
Thus, the side lengths $3$ cm, $4$ cm , $7$ cm will not form a triangle.
Hence, option D is correct.