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Triangle inequality related to lines and triangles - class-IX

Description: triangle inequality related to lines and triangles
Number of Questions: 30
Created by:
Tags: geometry triangles congruency of triangles geometric proof the triangle and its properties triangle and its properties construction of parallel lines and triangles triangles and their properties maths
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In any triangle, the side opposite to the larger (greater) angle is longer

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. True

  2. False

Show answer
Correct Option: A
Explanation:

A greater angle of a angle is opposite a greater side$.$ Let $ABC$ be a triangle in which angle $ABC$ is greater than angle $BCA;$ then side $AC$ is also greater than side $AB.$ For if it is no greater$,$ then $AC$ is either equal to $AB$ or less$.$

Hence$,$ option $(A)$ is always true$.$

For a triangle $ABC$, the true statement 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. ${ AC }^{ 2 }={ AB }^{ 2 }+{ BC }^{ 2 }$

  2. $AC=AB+BC$

  3. $AC>AB+BC$

  4. $AC<\,AB+BC$

Show answer
Correct Option: D
Explanation:

For any $\triangle ABC$, sum of two sides must be greater than the third side.
Hence, $AB + BC > AC$.

Two sides of a triangle are of lengths 5 cm and 1.5 cm, then the length of the third side of the triangle cannot 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. $\displaystyle3.6\,cm$

  2. $\displaystyle4.1\, cm$

  3. $\displaystyle3.8\, cm$

  4. $\displaystyle3.4\, cm$

Show answer
Correct Option: D
Explanation:

In a triangle, the difference between two sides should be less than the third side.
Hence,option D is correct $3.4\ cm$

Which of the following sets of side lengths 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. 4 m, 3 m, 11 m

  2. 7 mm, 4 mm, 4 mm

  3. 3 cm, 1.23 cm, 5 cm

  4. 3 m, 10 m, 8 m

Show answer
Correct Option: B
Explanation:

A triangle can be formed only if sum of any two sides is greater than the third side.

In option $C$ sum of any two sides taken a time is greater than the third side.
$7+4>4$
$4+4>7$
$4+7>4$
So option $C$ is correct.

Which is the smallest side in the following triangle?
$\displaystyle \angle P:\angle Q:\angle R=1:2:3$

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. $PQ$

  2. $QR$

  3. $PR$

  4. cannot be determined

Show answer
Correct Option: B
Explanation:

Given, $\angle P: \angle Q: \angle R=1:2:3$

By applying relationship between sides and angles of a triangle, if two sides of a triangle are unequal, the side opposite to smaller angle is smaller.
Since, $\angle P$ is smallest, so $QR$ is smallest.

It is not possible to construct a triangle with which of the following sides?

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. $8.3\ cm, 3.4\ cm, 6.1\ cm$

  2. $5.4\ cm, 2.3\ cm, 3.1\ cm$

  3. $6\ cm, 7\ cm, 10\ cm$

  4. $3\ cm, 5\ cm, 5\ cm$

Show answer
Correct Option: B
Explanation:

A triangle can be formed only if sum of any two sides is greater than the third side.

In option $C$
$2.3cm+3.1cm=5.4cm$
which is equal to the third side.
So a triangle can not be constructed.

The sides of a triangle (in cm) are given below: 

In which case, the construction of $\triangle $ is not possible?

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. 8, 7, 3

  2. 8, 6, 4

  3. 8, 4, 4

  4. 7, 6, 5

Show answer
Correct Option: C
Explanation:

A triangle can be formed if the sum of any two sides of triangle is greater then the third side

In option $C$ , sum of two sides is equal to third side.
So triangle can not be formed.
Option $C$ is correct.

In $\Delta PQR, \angle P = 60^{\circ}$ and $\angle Q = 50^{\circ}$. Which side of the triangle is the longest ?

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. PQ

  2. QR

  3. PR

  4. None

Show answer
Correct Option: A
Explanation:

Using angle sum property of triangle 

$\angle P+\angle Q+\angle R={ 180 }^{ \circ  }\ \Rightarrow { 60 }^{ \circ  }+{ 50 }^{ \circ  }+\angle R={ 180 }^{ \circ  }\ \Rightarrow \angle R={ 180 }^{ \circ  }-{ 110 }^{ \circ  }={ 70 }^{ \circ  }$

So $\angle R$ is the largest angle and side opposite to largest angle is the longest side.
$\therefore PQ$ is the longest side.

It is not possible to construct a triangle when its sides are :

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. 8.3 cm, 3.4 cm, 6.1 cm

  2. 5.4 cm, 2.3 cm, 3.1 cm

  3. 6 cm, 7 cm, 10 cm

  4. 3 cm, 5 cm, 5 cm

Show answer
Correct Option: B
Explanation:

For forming a triangle sum of any two sides must be greater than the third side.

In option $B$
$2.3cm+3.1cm=5.4cm$
which is equal to the third side.
So a triangle can not be formed.
Option $B$ is correct.

In $\Delta ABC, \angle B = 30^{\circ}, \angle C = 80^{\circ}$ and $\angle A = 70^{\circ}$ 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. $AB > BC < AC$

  2. $AB < BC > AC$

  3. $AB > BC > AC$

  4. $AB < BC < AC$

Show answer
Correct Option: C
Explanation:

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

Here $\angle C$ is largest and side opposite to it is $AB$
$\therefore AB$ is the longest side.
Then comes  $\angle A$ and side opposite to it is $BC$.
$\therefore BC$ is second longest.
Then comes $\angle B$ and side opposite to it is $AC$
So it is the smallest side.
So the decreasing order of sides is 
$AB>BC>AC$

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

Show answer
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

Show answer
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. $\angle A = \angle B$

Show answer
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

Show answer
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

Show answer
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

Show answer
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. $\displaystyle \angle P$ < $\displaystyle \angle Q$>$\displaystyle \angle R$

Show answer
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 $

Show answer
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. $0$

Show answer
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

Show answer
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. 

Which is the greatest side in the following triangle?
$\displaystyle \angle A:\angle B:\angle C=4:5:6$

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 be determined

Show answer
Correct Option: A
Explanation:

Let $\angle A: \angle B: \angle C=4x:5x:6x$
$\therefore 4x+5x+6x=180$
$\therefore 15x=180$
$\therefore x=12$
Largest angle $=\angle C=6x=6\times 12=72$
Side opposite to greatest angle has greatest length. 
According to the given ratio, $\displaystyle \angle C$ is the greatest angle and thus$,$ $AB$ is the greatest side.

The length of two sides of a triangle are $20 $ mm and $29 $ mm. Which of the following can be the value of third side to form the 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. $6 $ mm

  2. $7 $ mm

  3. $23 $ mm

  4. $8 $ mm

Show answer
Correct Option: C
Explanation:

We know that $(29-20) $ mm should smaller than the third side. 

Thus, the third side is greater than $9 $ mm.
Also, third side should be less than sum of $20$ and $29 $ mm  i.e. $49
$ mm.
Thus, $23 $ mm can be the length of third side to form a triangle.

The lengths of two sides of a triangle are $7 $ cm and $10 $ cm. What is the possible value range of the third side?

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. $3 $ cm $<$ third side $< 10 $ cm

  2. $7 $ cm $<$ third side $< 10 $ cm

  3. $3 $ cm $<$ third side $< 17 $ cm

  4. $7 $ cm $<$ third side

Show answer
Correct Option: C
Explanation:

We know that:
(i) The sum of lengths of any two sides of a triangle is greater than the third side. Thus, we know that $(7 + 10) $ cm is greater than the third side.
Therefore, third side is less than $17 $ cm.
(ii) The difference of lengths of any two sides of triangle is smaller than the third side. Thus $(10 - 7) \ cm$ is smaller than the third side.
Therefore, third side is greater than $3 $ cm
Thus, $3 $ cm $<$ third side $< 17 $ cm.

The lengths of two sides of a triangle are $3 $ cm and $4 $ cm. Which of the following, can be the length of third side to 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. $0.5 $ cm

  2. $5 $ cm

  3. $8 $ cm

  4. $10$ cm

Show answer
Correct Option: B
Explanation:

(I) We know that $(3 + 4) $ cm is greater than third side.
Thus, the third side is smaller than $7$ cm
(ii) we know that $(4 - 3) $ cm is smaller than third side .
Thus, the third side is greater than $1 $cm.
Therefore, $1 $ cm < third side $< 7 $ cm.

Thus, $5  $ cm can be the length of third side for a triangle. 

Find all possible lengths of the third side, if sides of a triangle have $3$ and $9$.

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. $6 < x < 12$

  2. $5 < x < 12$

  3. $6 < x < 10$

  4. $6 < x < 11$

Show answer
Correct Option: A
Explanation:

The Triangle Inequality theorem states that the sum of any $2$ sides of a triangle must be greater than the measure of the third side.
So, difference of two sides $< x <$ sum of two sides, will give you the possible length of a triangle.
Therefore, $9 - 3 < x < 9 + 3$
$6 < x < 12$ is the possible length of the third side of a triangle.
For checking the possible length: Take $3, 9, 7$
$3 + 9 > 7 (a + b > c)$
$9 + 7 > 3 (b + c > a)$
$3 + 7 > 9 (a + c > b)$
Which satisfy the triangle inequality theorem.

The construction of a triangle $ABC$, given that $BC =$ $6$ cm, $B =$ $45 ^{\circ}$ is not possible when difference of $AB$ and $AC$ 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. $6.9$ cm

  2. $5.2$ cm

  3. $5.0$ cm

  4. $4.0$ cm

Show answer
Correct Option: A
Explanation:

According to the theorem of inequalities, the sum of any two sides of the triangle is greater than the third side.

Therefore, $AC+BC>AB$
$\Rightarrow BC>AB-AC$
Therefore, only the first option that is $6.9$ cm does not satisfy the above equation. Rest all the options satisfy the equation.

In triangle ABC, (b+c) cos A+(c+a)cos B+(a+b)cos C 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. $0$

  2. $1$

  3. $a+b+c$

  4. $2(a+b+c)$

Show answer
Correct Option: C
Explanation:

$(b+c) \cos A+(c+a)\cos B+(a+b)\cos C$


$\Rightarrow$  $b\cos A+c\cos A+c\cos B+a\cos B+a\cos C+b\cos C$

$\Rightarrow$  $(b\cos C+c\cos B)+(c\cos A+a\cos C)+(a\cos B+b\cos A)$  ----( 1 )
Using projection formula,
$a=(b\cos C+c\cos B)$
$b=(c\cos A+a\cos C)$
$c=(a\cos B+b\cos A)$
Substituting above values in ( 1 ) we get,
$\Rightarrow$  $a+b+c$
$\therefore$   $(b+c) \cos A+(c+a)\cos B+(a+b)\cos C=a+b+c$

Find all possible lengths of the third side, if sides of a triangle have $2$ and $5$.

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. $2 < x < 7$

  2. $3 > x < 7$

  3. $3 < x > 7$

  4. $3 < x < 7$

Show answer
Correct Option: D
Explanation:

The Triangle Inequality theorem states that the sum of any $2$ sides of a triangle must be greater than the measure of the third side.
So, difference of two sides $< x <$ sum of two sides, will give you the possible length of a triangle.
Therefore, $5 - 2 < x < 5 + 2$
$3 < x < 7$ is the possible length of the third side of a triangle.
For checking the possible length: Take $2, 5, 4$
$2 + 5 > 4 (a + b > c)$
$5 + 4 > 2 (b + c > a)$
$2 + 4 > 5 (a + c > b)$
Hence, the above condition satisfied the triangle inequality theorem.

A triangle has side lengths of $6$ inches and $9$ inches. If the third side is an integer, calculate the minimum possible perimeter of the triangle (in inches).

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. $15$

  3. $8$

  4. $19$

  5. $29$

Show answer
Correct Option: D
Explanation:

Let the third side be $x$.
Sum of any two sides of a triangle is greater than the third side. 

Hence, $6+x>9$ or $x>3$ and $6+9>x$ or $x<15$.
Therefore, $x\epsilon (3,15)$
Hence, the minimum possible integral value of $x$ is $4$. 
Thus the minimum possible length of the third side is $4$. 
Hence, the minimum possible perimeter is $4+6+9=19$ units.

Which statement is true about the difference of any two sides of 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. It is greater than the third side

  2. It is zero

  3. It is lesser than the third side

  4. It is lesser than zero

Show answer
Correct Option: C
Explanation:

Let $a,b,c$ be the sides of triangle.

For constructing a triangle sum of any two sides must be greater than third side
$\Rightarrow a+b>c$
$\Rightarrow a>c-b$
$\Rightarrow c-b<a.......(i)$
Also $a+c>b$
$\Rightarrow  c>a-b$
$\Rightarrow a-b>c.....(ii)$
Also $b+c>a$
$\Rightarrow c>a-b$
$\Rightarrow a-b>c.......(iii)$
From $(i),(ii)$ and $(iii)$ it is clear that difference of any two sides is greater than the third side.
So option $C$ is correct.

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