Tag: inequalities in triangle

Questions Related to inequalities in triangle

In triangle ABC, (a +b -c )(b+c -a)(c+a-b)-abc is always ;

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. non negative

  2. non positive

  3. negative

  4. positive

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Correct Option: A

In a triangle $ABC, \angle C=90^{o}, a=3, b=4$ and $D$ is a point on $AB$, so that $\angle BCD=30^{o}$. Then the length of $CD$ is 

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. $\dfrac{18-24\sqrt{3}}{25}$

  2. $\dfrac{18+24\sqrt{3}}{25}$

  3. $\dfrac{8-24\sqrt{3}}{25}$

  4. $None\ of\ these$

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Correct Option: D
Explanation:
Area of $\triangle BCD$ + Area of $\triangle ACD$ = Area of $\triangle ABC$

$\dfrac{1}{2}\cdot 3CD\cdot \sin 30+\dfrac{1}{2}\cdot 4CD\cdot \sin 60=\dfrac{1}{2}3\times 4$

$3CD\times \dfrac{1}{2}+4CD \times \dfrac{\sqrt{3}}{2}=12$

$\dfrac{CD}{2}(3+4\sqrt{3})=12$

$\therefore CD=\dfrac{24}{3+4\sqrt{3}}$

In a triangle $ABC, \angle ABC=50^{o}, \angle BAC=30^{o}$, then the shortest sides is 

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. $AB$

  2. $BC$

  3. $CA$

  4. $None$

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Correct Option: B
Explanation:
The side opposite to the largest angle is the longest side of the triangle and the side opposite to the smallest angle is the shortest side of the triangle.
By angle sum property,$ \angle{A}+\angle{B}+\angle{C}={180}^{\circ}$
$\Rightarrow {30}^{\circ}+{50}^{\circ}+\angle{C}={180}^{\circ}$
$\Rightarrow \angle{C}={180}^{\circ}-{80}^{\circ}={100}^{\circ}$
The side opposite to the smallest angle $\left({30}^{\circ}\right)$ is the side $BC$

If two sides of a triangle are 8 cm and 13 cm, then the length of the third side is between a cm and b cm. Find the values of a and b such  that a is less than b.

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. a = 2 cm and b =9 cm

  2. a = 7 cm and b = 12 cm

  3. a = 5 cm and b = 21 cm

  4. a = 8 cm and b = 24 cm

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Correct Option: C
Explanation:

$If  \ two \  sides  \ of \  a \  triangle \  are \  8 cm \  and \  13 cm.$
$Then  \ the  \ length \  of  \  third \  side  \  is$
$more \  than \  13-8 \ =  5cm$
$and  \ less \  than \  13+8  =  21cm$
$a = 5cm , b = 21cm$

Two sides of a triangle are $7$ and $10$ units, which of the following length can be the length of the third side?

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. $19$ units

  2. $17$ units

  3. $13$ units

  4. $3$ units

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Correct Option: C
Explanation:
The sum of any two sides is greater than the third side.
and difference between two sides should be lesser than the third side.
For option $A,$ $7+10\ngtr 19$ 
For option $B,$ $7+10\ngtr 17$ 
For option $C,$ $7+10 > 13$ 
Also, $ |10-7| < 13$
For option $D,$ $7+10 > 3$ 
but $|7-10| \nless 3$
Only, $C$ is correct.

If AB > AC and length of median from C is 4 units, then the length of median from B can be

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. 2

  2. 3

  3. 4

  4. 5

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Correct Option: D
Explanation:

This can be taken as a fact that the median corresponding to the longer side has the larger median.

Or you can even take a rough triangle to deduce the result.

If $k$ is an integer and $2 < k < 7$, for how many different values of $k$ is there a triangle with sides of lengths $2, 7$, and $k$?

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. One

  2. Two

  3. Three

  4. Four

  5. Five

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Correct Option: A
Explanation:
In a triangle, the sum of the smaller two sides must be larger than the largest side.
For $k$ values $3, 4, 5$, and $6$, the only triangle possible is $2, 7$, and $k = 6$ because only $2 + 6 > 7$. For $k$ values $3, 4$, and $5$, the sum of the smaller two sides is not larger than the third side; thus, $6$ is the only possible value of $k$ that satisfies the conditions.
The correct answer is A.

All equilateral triangles are ____.

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. congruent

  2. will be congruent if their length of sides are equal

  3. never congruent

  4. none of these

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Correct Option: B
Explanation:

Equilateral triangles have all the three angles equal to $\displaystyle { 60 }^{ o }$. 

But the length of sides can be different for different equilateral triangles. 
Thus if the length is also fixed, then all the equilateral triangles having sides lengths same are congruent.

Find the length of the third side of the triangle inequality, if sides of a triangle have $a = 4$ and $b = 8$.

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. $c = 10$

  2. $c = 2$

  3. $c = 3$

  4. $c = 4$

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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.
Take option A: $a = 4, b = 8, c = 10$
$4 + 8 > 10 (a + b > c)$
$8 + 10 > 4 (b + c > a)$
$4 + 10 < 8 (a + c > b)  $
Therefore, the third side, $c = 10$ will satisfy the triangle inequality
Similarly check for option 2: $c = 2$
$4 + 8 > 2 (a + b > c)$
$8 + 2 > 10 (b + c = a)$
$4 + 2 < 8 (a + c < b)  $
Hence, the second option will not satisfying the triangle inequality.
Similarly check for option $3$ and $4$.
Here, the third side of the triangle inequality, $c = 10$.

In triangle $PQR$, an exterior angle at $A$ measures $160^o$, and $\angle$ $Q = 70 ^o$. Which is the longest side of the triangle?

maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

  1. $\overline{PR}$

  2. $\overline{PA}$

  3. $\overline{PQ}$

  4. $\overline{QR}$

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Correct Option: C
Explanation:

Exterior angle $A = 180 -$ $\angle$ $P = 180-160$
$\angle$ $P = 20$
$\angle$ $Q = 70$
Interior angle $= 20 + 70 +$ $\angle$ $R = 180$
$\angle$ $R = 90$
In a triangle inequality theorem,the largest side is across from the longest angle.
So, $90^o$ is the longest angle in the triangle, $\overline{PQ}$, across from it, is the largest side.
Therefore, $\overline{PQ}$ is the largest side.