Tag: maths

Questions Related to maths

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

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

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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.