Tag: congruence and inequalities of triangles

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Questions Related to congruence and inequalities of triangles

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

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

  1. non negative

  2. non positive

  3. negative

  4. positive
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let a, b, c be sides of a triangle. By the triangle inequality, (a+b-c) > 0, (b+c-a) > 0, and (c+a-b) > 0. The expression (a+b-c)(b+c-a)(c+a-b) - abc is known to be non-negative for all triangles.

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

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 

  1. $\dfrac{18-24\sqrt{3}}{25}$
  2. $\dfrac{18+24\sqrt{3}}{25}$
  3. $\dfrac{8-24\sqrt{3}}{25}$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Area of $\triangle BCD$ + Area of $\triangle ACD$ = Area of $\triangle ABC$

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

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

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

$\therefore CD=\dfrac{24}{3+4\sqrt{3}}$
Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

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

  1. $AB$
  2. $BC$
  3. $CA$
  4. $None$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $BC$

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

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.

  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
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

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

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

  1. $19$ units
  2. $17$ units
  3. $13$ units
  4. $3$ units
Reveal answer Fill a bubble to check yourself
C Correct answer
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.
Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

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 $k$?

  1. One

  2. Two

  3. Three

  4. Four

  5. Five
Reveal answer Fill a bubble to check yourself
A Correct answer
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.
Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

All equilateral triangles are ____.

  1. congruent

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

  3. never congruent

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

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

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

  1. $c = 10$
  2. $c = 2$
  3. $c = 3$
  4. $c = 4$
Reveal answer Fill a bubble to check yourself
A Correct answer
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, $c = 10$.

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

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

  1. $\overline{PR}$
  2. $\overline{PA}$
  3. $\overline{PQ}$
  4. $\overline{QR}$
Reveal answer Fill a bubble to check yourself
C Correct answer
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.