Mathematics · Quantitative Aptitude

Geometry of Triangles and Angles

758 Questions

Triangle and angle geometry covers angle sums, properties of equilateral shapes, and right triangles. These principles form the basis of advanced quantitative aptitude sections. Practicing spatial problems helps secure points in exams.

Triangle angle sumsPythagorean theoremProperties of equilateral trianglesComplementary and supplementary anglesRight triangle formulas

Geometry of Triangles and Angles Questions

Multiple choice

ABC is a right triangle in which angle B = 90°. The biggest circle possible is inscribed in the triangle. If AB = 8 cm and BC = 6 m, find the radius r of the incircle.

  1. 4 cm

  2. 1 cm

  3. 2 cm

  4. 0.5 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Area of right $\triangle$ABC = 1/2 x AB x BC= 1/2 x (6 x 8)cm2 = 24 cm2     AC2 = AB2 + BC2                             64 cm2 + 36 cm2 = 100 cm2. AC = 10 cm.  Also, area ($\triangle$ABC) = ar ($\triangle$OBC) + ar ($\triangle$OCA) + ar ($\triangle$OAB)= 1/2 (BC x r) + 1/2 (AC x r) + 1/2 (AB x r)= 1/2 r (6 + 10 +8) cm = 12 r cm 12 r cm = 24 cm2 $\Rightarrow$ r = 2 cm

Multiple choice maths measures and the circle surface area and volume of sphere surface area of a prism surface area of a prism and a pyramid

The base of right prism is a triangle whose perimeter is 28 cm and the inradius of the triangle is 4 cm. If the volume of the prism is 366 cc, then its height is 

  1. 6.54 cm

  2. 8 cm

  3. 4 cm

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Perimeter of triangle is $2{s}=28\ cm\implies s=14$

Inradius of triangle $r=4\implies \Delta=r.s=56$
Volume of prism $=366$ cc
$\implies $ Area of triangle $\times $ height $=366$
Height $=\dfrac{366}{56}=6.54$

Multiple choice 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

In any triangle, the side opposite to the larger (greater) angle is longer

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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$.$

Multiple choice 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

The sides of a triangle (in cm) are given below: 

In which case, the construction of $\triangle $ is not possible?

  1. 8, 7, 3

  2. 8, 6, 4

  3. 8, 4, 4

  4. 7, 6, 5
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

Multiple choice 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

In $\Delta ABC, \angle B = 30^{\circ}, \angle C = 80^{\circ}$ and $\angle A = 70^{\circ}$ then,

  1. $AB > BC < AC$
  2. $AB < BC > AC$
  3. $AB > BC > AC$
  4. $AB < BC < AC$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 
$AB>BC>AC$

Multiple choice 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

In $\Delta ABC$, if $\angle A = 50^{\circ}$ and $\angle B = 60^{\circ}$, then the greatest side is :

  1. AB

  2. BC

  3. AC

  4. Cannot say
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using angle sum property of triangle

$\angle A+\angle B+\angle C={ 180 }^{ \circ  }\ \Rightarrow { 50 }^{ \circ  }+{ 60 }^{ \circ  }+\angle C={ 180 }^{ \circ  }\ \Rightarrow \angle C={ 180 }^{ \circ  }-{ 110 }^{ \circ  }={ 70 }^{ \circ  }$
Side opposite to the largest angle is the longest side.
Here $\angle C$ is largest and side opposite to it is $AB$
So $AB$ is the longest side.

Multiple choice 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

In $\Delta ABC$, if $\angle A = 35^{\circ}$ and $\angle B = 65^{\circ}$, then the longest side of the triangle is :

  1. AC

  2. AB

  3. BC

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

Using angle sum property of triangle

$\angle A+\angle B+\angle C={ 180 }^{ \circ  }\ \Rightarrow { 35 }^{ \circ  }+{ 65 }^{ \circ  }+\angle C={ 180 }^{ \circ  }\ \Rightarrow \angle C={ 180 }^{ \circ  }-{ 100 }^{ \circ  }={ 80 }^{ \circ  }$
Side opposite to largest angle is the longest side of any triangle.
Here $\angle C$ is the largest angle and side opposite to it is $AB$
So $AB$ is the longest side.

Multiple choice 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

In $\Delta ABC$, if AB $>$ BC then :

  1. $\angle C < \angle A$
  2. $\angle C = \angle A$
  3. $\angle C > \angle A$
  4. $\angle A = \angle B$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In any triangle angle opposite to the longest side is largest.

Angle opposite to $AB$ is $\angle C$ and angle opposite to $BC$ is $A$
Here $AB>AC$
$\Rightarrow \angle C>\angle A$
So option $C$ is correct.

Multiple choice 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

In $\Delta ABC, \angle A=100^{\circ}, \angle B=30^{\circ}$ and $\angle C= 50^{\circ}$,then

  1. $AB>AC$
  2. $AB=AC$
  3. $AB<AC$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
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. 
Multiple choice 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

Which is the greatest side in the following triangle?
$\displaystyle \angle A:\angle B:\angle C=4:5:6$

  1. $AB$
  2. $BC$
  3. $AC$

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice 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

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:

  1. $6.9$ cm
  2. $5.2$ cm
  3. $5.0$ cm
  4. $4.0$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice 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

In triangle ABC, (b+c) cos A+(c+a)cos B+(a+b)cos C is equal to

  1. $0$
  2. $1$
  3. $a+b+c$
  4. $2(a+b+c)$
Reveal answer Fill a bubble to check yourself
C Correct answer
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,
$\Rightarrow$  $a+b+c$
$\therefore$   $(b+c) \cos A+(c+a)\cos B+(a+b)\cos C=a+b+c$

Multiple choice 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

Which statement is true about the difference of any 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
Reveal answer Fill a bubble to check yourself
C Correct answer
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.