Tag: triangle inequality

Questions Related to triangle inequality

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.

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.