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 maths constructions mid-point formula midpoints division of a line segment

If $(2, 3), (-4, 5), (1, -2)$ are the midpoints of the sides $\vec{BC}, \vec{CA}, \vec{AB}$ of $\triangle ABC$, then the equation of $\vec{AB}$ is 

  1. $3x-y-5=0$
  2. $x+3y+5=0$
  3. $x+3y-11=0$
  4. $3x-y+17=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let vertices be A(x1, y1), B(x2, y2), C(x3, y3). Using the given midpoints, set up midpoint equations to find the coordinates of vertices A and B. Once vertices A and B are determined, find the equation of the line passing through them, which results in x + 3y + 5 = 0.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

If an triangle ABC, A = {1, 10}, circumference = $\left( -\dfrac { 1 }{ 3 } ,\dfrac { 2 }{ 3 }  \right) $ and orthocenter = $\left( \dfrac { 11 }{ 3 } ,\dfrac { 4 }{ 3 }  \right) $ then the co-ordinate of mid-point of side opposite to A is ________.

  1. (1, 11/3)

  2. (1, 5)

  3. (1, -3)

  4. (1, 6)
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths constructions mid-point formula midpoints division of a line segment

If $(-2,3), (4,-3), (4,5)$ are mid-points of the sides of a triangle, find the coordinates of the centroid of the triangle formed by these mid-points.

  1. $\left (3,\dfrac43 \right )$
  2. $\left (2,\dfrac43 \right )$
  3. $\left (2,\dfrac53 \right )$
  4. $\left (3,\dfrac53\right )$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the given vertices of a triangle be A$(-2,3)$ and B $(4,-3)$ let the third vertex be C $(4,5)$.


Let Centroid be $G= \left [\dfrac {x _1 + x _2 + x _3}{3} , \dfrac{y _1 +y _2 + y _3}{3}\right ]$


$\Rightarrow G$ = $\left (\dfrac{-2 +4+4 }{3} ,\dfrac{3-3+5}{3}\right )$

$\Rightarrow G$ = $\left (\dfrac{6 }{3} , \dfrac{5}{3}\right )$

$\Rightarrow G$ = $\left (2 , \dfrac {5}{3}\right )$.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Find the third vertex of a triangle, if two of its vertices are $(-3,1), (0,-2)$ and centroid is at the origin.

  1. $(3,4)$
  2. $(2,1)$
  3. $(3,2)$
  4. $(3,1)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the third vertex be $(a,b)\equiv(x _1,y _1),(-3,1)\equiv(x _2,y _2), (0,-2)\equiv(x _3,y _3)$


Centroid of triangle is $(0,0)$

$G =\left [\dfrac {x _1 + x _2 + x _3}{3} , \dfrac{y _1 +y _2 + y _3}{3}\right ]$

$\Rightarrow \left( \dfrac { a-3+0 }{ 3 } ,\dfrac { b+1-2 }{ 3 }  \right) =(0,0)$

$\Rightarrow\dfrac{a-3}3=0$ and $\dfrac{b-1}3=0$

$\Rightarrow a=3,b=1$

So the third vertex is $(3,1)$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

$A\equiv(0, b), B\equiv(0, 0) $ and $C\equiv(a, 0)$ are the vertices of $\triangle ABC. D, E, F$ are the mid-points of the sides $BC, CA $ and $AB $ respectively. If  $a^{2}+ b^{2} = 20$ then

  1. $(AD)^{2}=9$
  2. $(BE)^{2}=4$
  3. $(AD)^{2}+(CF)^{2}=25$
  4. $(AD)^{2}+(CF)^{2}=(BE)^{2}$
Reveal answer Fill a bubble to check yourself
C,D Correct answer
Explanation

As $A\equiv\left( 0,b \right) ,B\equiv\left( 0,0 \right) $ and $C\equiv\left( a,0 \right) $ 
Then $D\equiv\left( \cfrac { a }{ 2 } ,0 \right) ,E\equiv\left( \cfrac { a }{ 2 } ,\cfrac { b }{ 2 }  \right) $ and $F\equiv\left( 0,\cfrac { b }{ 2 }  \right) $
Using $a^{ 2 }+b^{ 2 }=20$, we have
${ \left( AD \right)  }^{ 2 }={ \left( \cfrac { a }{ 2 } -0 \right)  }^{ 2 }+{ \left( 0-b \right)  }^{ 2 }=\cfrac { { a }^{ 2 } }{ 4 } +{ b }^{ 2 }\ Similary\,\, {(CF)}^2\ { \left( AD \right)  }^{ 2 }+{ \left( CF \right)  }^{ 2 }={ \left( \cfrac { a }{ 2 } -0 \right)  }^{ 2 }+{ \left( 0-b \right)  }^{ 2 }+{ \left( 0-a \right)  }^{ 2 }+{ \left( \cfrac { b }{ 2 } -0 \right)  }^{ 2 }\ =\cfrac { 5\left( { a }^{ 2 }+{ b }^{ 2 } \right)  }{ 4 } =25={ \left( BE \right)  }^{ 2 }$

Multiple choice maths calculations and mental strategies 1 equations from statements forming equations from statements writing mathematical statements

Which of the equation satisfy the given statement ?
"Perimeter of an equilateral triangle is three times its side l" 

  1. $l \times l \times l$
  2. $3 \times 3 \times 3$
  3. $3 \times l$
  4. $3 + l$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
An equilateral triangle is a triangle in which all three sides are equal.Let $l$ be the length of the side of a triangle.
Thus, perimeter is equal to sum of all its sides$3\times side=3\times l$
Multiple choice

Sum of angles in a traingle.

  1. 90

  2. 120

  3. 180

  4. 360
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The sum of the interior angles of any triangle in Euclidean geometry is always 180 degrees.

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

In a square $ABCD$, the bisector of the angle $BAC$ cut $BD$ at $X$ and $BC$ at $Y$ then triangles $ACY, ABX$ are similar.

  1. True

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

In square ABCD, angle BAC = 45 degrees. The bisector of BAC makes angles of 22.5 degrees. Through geometric properties and angle chasing, it can be shown that triangles ACY and ABX are similar.

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

Consider the following statements:
(1) If three sides of triangle are equal to three sides of another triangle, then the triangles are congruent.
(2) If three angles of a triangle are respectively equal to three angles of another triangle, then the two triangles are congruent.

Of these statements,

  1. $(1)$ is correct and $(2)$ is false
  2. Both $(1)$ and $(2)$ are false
  3. Both $(1)$ and $(2)$ are correct
  4. $(1)$ is false and $(2)$ is correct
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If three sides of triangle are equal to other three sides of the triangle, then the two triangles are congruent by SSS rule.

If three angles of the triangle are equal to other three sides of the triangle, then the two triangles are similar by AAA rule but not congruent.

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

If in trianges $ABC$ and $DEF$, $\cfrac{AB}{DE}=\cfrac{BC}{FD}$, then they will be similar, when:

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

In $\triangle ABC$ and $\triangle DEF$,
$\dfrac{AB}{DE} = \dfrac{BC}{FD}$ (Given)

The angle between these sides are $\angle B$ and $\angle D$. Thus, If the containing angles are equal. The triangles will be similar..

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

In a $\triangle ABC$, $BC=AB$ and $\angle B={ 80 }^{ 0 }$. Then $\angle A$ is equal to?

  1. ${ 80 }^{ 0 }$
  2. ${ 40 }^{ 0 }$
  3. ${ 50 }^{ 0 }$
  4. ${ 100 }^{ 0 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given: $BC = AB$, $\angle B = 80^{\circ}$
Since, $BC = AB$
$\angle A = \angle C = x$ (Angles opposite to equal sides are equal)
Sum of angles of a triangle = 180
$\angle A + \angle B + \angle C = 180$
$x + 80 + x = 180$
$2x = 100$
$x = 50^{\circ}$
Thus, $\angle A = 50^{\circ}$

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

If the areas of two similar triangles are equal then the triangles :

  1. are congruent

  2. have equal length of corresponding sides

  3. (A) and (B)

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

Triangles are similar if they have the same shape, but not necessarily the same size.   It is same as keeping its basic shape but either "zooming in" or out making the triangle bigger or smaller .

For triangles to be congruent if one triangle slides over the other , rotate them, and flip them over in various ways so they will exactly fit over each other.

So, if the triangles are similar, i.e. they are "zoomed-in" or "zoomed-out" versions of each other, and if they have equal area, then they are congruent, and hence have equal corresponding sides.