Tag: triangle inequality

Questions Related to triangle inequality

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

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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. $AC<\,AB+BC$

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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. $\displaystyle3.4\, cm$

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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

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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

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

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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

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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

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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

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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. $AB < BC < AC$

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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 
$AB>BC>AC$