Tag: sum of the lengths of two sides of a triangle

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Questions Related to sum of the lengths of two sides of a triangle

Multiple choice 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

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

  1. AB

  2. BC

  3. AC

  4. Cannot say
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice 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

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

  1. AC

  2. AB

  3. BC

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

Multiple choice 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

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

  1. $\angle C < \angle A$
  2. $\angle C = \angle A$
  3. $\angle C > \angle A$
  4. $\angle A = \angle B$
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

Multiple choice 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

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

  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
Reveal answer Fill a bubble to check yourself
A Correct answer
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. 

Multiple choice 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

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

  1. $AB>AC$
  2. $AB=AC$
  3. $AB<AC$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
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. 
Multiple choice 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

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.......

  1. 14 cm

  2. 15 cm

  3. 21 cm

  4. 27 cm
Reveal answer Fill a bubble to check yourself
D Correct answer
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 $=7+7+13=27\ cm$
Multiple choice 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

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:

  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. $\displaystyle \angle P$ < $\displaystyle \angle Q$>$\displaystyle \angle R$
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

Multiple choice 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

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

  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 $
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $\angle P>\angle Q>\angle R$.

Multiple choice 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

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

  1. $4$
  2. $2$
  3. $1$
  4. $0$
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

Multiple choice 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

Which of the following sets of side lengths will not form 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
Reveal answer Fill a bubble to check yourself
D Correct answer
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.