Mathematics · Quantitative Aptitude

Geometry of Triangles and Angles

846 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 pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

Mark the correct alternative of the following.
In a right triangle, one of the acute angles is four times the other. Its measure is?

  1. $68^o$
  2. $84^o$
  3. $80^o$
  4. $72^o$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In the right-angled triangle the sum of the other two angles is $90^o$.

Let one acute angle is $x$, then the other angle is $4x$. [ Given]
Then we get,
$4x+x=90^o$
or, $5x=90^o$
or, $x=18^o$.
So the measure of that angle is $4\times 18^o=72^o$.

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

If the sides of a triangle are in the ratio $1\, :\, \sqrt2\, :\, 1$, then the triangle is:

  1. an equilateral triangle

  2. an isosceles triangle

  3. a right angled triangle

  4. a right angled isosceles triangle
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given ratio of sides of the triangle, $1 : \sqrt{2} : 1$
Let the triangle be $ABC$ and sides be
$AB = x$
$BC = x $
$AC = \sqrt{2}x$

Clearly, $AC^2 = BC^2 + AB^2$
Hence, by converse of Pythagoras theorem, $ABC$ is a right-angled isosceles triangle.

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

In a $\Delta$ $ABC, AD = 3, BC = 2, AB = 1$, find the value of $AC$. (Use Apollonius theorem).

  1. $2.35$
  2. $3.42$
  3. $4.35$
  4. $5.61$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

According to the Apollonius theorem, 
$AB^{2}+AC^{2}= 2[AD^{2}+\dfrac {BC}{2}^{2}]$
$1^{2}+AC^{2}= 2[3^{2}+\dfrac{2}{2}^{2}]$
$1+AC^{2}= 2[9 + 1]$
$AC^{2}=20 -1$
$AC^{2}= 19$
$AC = \sqrt{19}$
$AC = 4.35$

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

In a $\Delta$ $ABC, AC = 6, BC = 2, AB = 4$, find the value of $AD$. (Use Apollonius theorem).

  1. 5

  2. 4

  3. 3

  4. 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

According to the Apollonius theorem, 
$AB^{2}+AC^{2}= 2\left [AD^{2}+\dfrac{BC}{2}^{2}\right]$
$4^{2}+6^{2}= 2\left [AD^{2}+\dfrac{2}{2}^{2}\right]$
$16+36= 2[AD^{2} + 1]$
$2AD^{2}=52 -2$
$2AD^{2}= 50$
$AD^{2} = \dfrac{50}{2}$
$AD^{2} = 25$
$AD = 5$

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

In a $\Delta$ $ABC, AC = 8, BC = 2, AB = 6$, find the value of $AD$. (Use Apollonius theorem).

  1. 3

  2. 5

  3. 7

  4. 9
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

According to the Apollonius theorem, 
$AB^{2}+AC^{2}= 2\left [AD^{2}+\dfrac{BC}{2}^{2}\right]$
$6^{2}+8^{2}= 2\left [AD^{2}+\dfrac{2}{2}^{2}\right]$
$36+64= 2[AD^{2} + 1]$
$2AD^{2}=100 -2$
$2AD^{2}= 98$
$AD^{2} = \dfrac{98}{2}$
$AD^{2} = 49$
$AD = 7$

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

In a $\Delta$ $ABC, AC = 4, BC = 2, AB = 6$, find the value of $AD$. (Use Apollonius theorem).

  1. 2

  2. 3

  3. 4

  4. 5
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

According to the Apollonius theorem, 
$AB^{2}+AC^{2}= 2\left [AD^{2}+\dfrac{BC}{2}^{2}\right]$
$6^{2}+4^{2}= 2\left [AD^{2}+\dfrac{2}{2}^{2}\right]$
$36+16= 2[AD^{2} + 1]$
$2AD^{2}=52 -2$
$2AD^{2}= 50$
$AD^{2} = \dfrac{50}{2}$
$AD^{2} = 25$
$AD = 5$

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

In any triangle, the sum of the squares on any two sides is equal to twice the square on half the third side together with twice the square on the median which bisects the third side is called ______ theorem.

  1. Pythagoras

  2. Apollinius

  3. Stewart

  4. Ceva's
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In any triangle, the sum of the squares on any two sides is equal to twice the square on half the third side together with twice the square on the median which bisects the third side is called Apollinius theorem.

Option $B$ is correct.

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

In a $\Delta$ $ABC, AC = 6, BC = 2, AB = 8$, find the value of $AD$. (Use Apollonius theorem).

  1. 5

  2. 6

  3. 7

  4. 8
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

According to the Apollonius theorem, 
$AB^{2}+AC^{2}= 2\left[AD^{2}+\dfrac{BC}{2}^{2}\right]$
$8^{2}+6^{2}= 2\left [AD^{2}+\dfrac{2}{2}^{2}\right]$
$64+36= 2[AD^{2} + 1]$
$2AD^{2}=100 -2$
$2AD^{2}= 98$
$AD^{2} = \dfrac{98}{2}$
$AD^{2} = 49$
$AD = 7$

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

Which one of the following formula is used to find apollinius theorem for isosceles triangle?

  1. $a^{2}+b^{2}=2m^{2}+\dfrac{c}{2}^{2}$
  2. $b^{2}=m^{2}+\dfrac{c}{4}^{2}$
  3. $b^{2}+b^{2}=2m^{2}+\dfrac{c}{2}^{2}$
  4. $a^{2}+b^{2}=2m^{2}+\dfrac{b}{2}^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In any triangle, the sum of the squares on any two sides is equal to twice the square on half the third side together with twice the square on the median which bisects the third side.
Apollinius theorem formula,
$a^{2}+b^{2}= 2[m^{2}+\dfrac{c}{2}^{2}]$
When the given triangle is isosceles, than $b = a$.
So, $b^{2}=m^{2}+\dfrac{c}{4}^{2}$ is the formula used for isosceles triangle.

Multiple choice maths pythagoras theorem similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

In a $\triangle ABC$, $AB= 4$ cm and $AC = 8$ cm. If M is the midpoint of BC and $AM = 3$ cm, then the length of $BC$ in cm is:

  1. ${2\sqrt{26}}$
  2. ${2\sqrt{31}}$
  3. ${\sqrt{31}}$
  4. ${\sqrt{26}}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given : In $\triangle ABC$, $AB=4$ cm and $AC=8$ cm.

M is the midpoint of BC and $AM=3$ cm
Using Apollonius theorem,
$AB^2+AC^2=2(AM^2+BM^2)$
$\implies$ $4^2+8^2=2(3^2+BM^2)$
$\implies$ $16+64=2(9+BM^2)$
$\implies$ $BM^2=31$
$\implies$ $BM=\sqrt{31}$.
$\because$ $BC=2BM$
$\therefore$ $BC=2\sqrt{31}$.

Multiple choice maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

Diagonals of trapezium $ABCD$ with $AB\parallel DC$ intersect each other at the point $O$. If $AB=2CD$, find the ratio of the areas of triangles $AOB$ and $COD$.

  1. $4:1$
  2. $4:3$
  3. $3:1$
  4. $4:7$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given: $AB \parallel CD$ and $AB = 2 CD$
In $\triangle OAB$ and $\triangle OCD$
$\angle AOB = \angle COD$ (Vertically opposite angles)
$\angle ODC = \angle OBA$ (Alternate angles)
$\angle OCD = \angle OAB$ (Alternate angles)
thus, $\triangle OAB \cong \triangle OCD$ (AAA rule)
$\dfrac{A(\triangle OAB)}{A(\triangle OCD)} = \dfrac{AB^2}{CD^2}$ (Similar triangle Property)
$\dfrac{A(\triangle OAB)}{A(\triangle OCD)} = \dfrac{(2 CD)^2}{CD^2}$
$\dfrac{A(\triangle OAB)}{A(\triangle OCD)} = 4 : 1$

Multiple choice maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

If ABCD is an isosceles trapezium, $\angle{C}$ is equal to 

  1. $\angle{B}$
  2. $\angle{A}$
  3. $90^\circ$
  4. $\angle{D}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If ABCD is a isosceles trapezium , $\angle C\quad will\quad be\quad equal\quad to\quad \angle D $
because angles on either side of the bases are same measure/size(congruent) in a isosceles trapezium.

Multiple choice maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

In a trapezium  $A B C D , A B |D C$  and  $D C = 2 A B .EF$  drawn parallel to  $AB$  cuts $AD$  in  $F$  and  $B C$  in E such that $ \dfrac { BE }{ EC } =\dfrac { 3 }{ 4 } $ Diagonal $DB $  intersects  $ EF$  at  $G$  then $7 \mathrm { FE } = 10 \mathrm { AB } .$

  1. True

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

Using the properties of similar triangles formed by the parallel lines in the trapezium, the ratio holds true.

Multiple choice maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

ABCD is a kite in which AB=AD and CB=CD, if $\angle ABD=40^{0},$ $then find \angle A+\angle C.$

  1. $220^{0}$
  2. $200^{0}$
  3. $180^{0}$
  4. $210^{0}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

we have,

$ABCD$ is a kite.

Then, Given that,

$ AB=AD\,\,\,\,\,.......\,\,\left( 1 \right) $

$ CB=CD\,\,\,\,\,.......\,\,\left( 2 \right) $

If $\angle ABD={{40}^{o}}$

Then, show Diagram,

$\angle ABD=\angle ADB={{40}^{o}}$

And

$ \angle ADB=\angle CBD={{40}^{o}}\,\,\left( \text{Alternate}\,\text{angle} \right) $

$ \angle ABD=\angle CDB={{40}^{o}}\,\,\,\left( \text{Alternate}\,\text{angle} \right) $

Then $\angle B+\angle D={{40}^{o}}+{{40}^{o}}+{{40}^{o}}+{{40}^{o}}={{160}^{o}}$

But, We know that,

In any quadrilateral

$ \angle A+\angle B+\angle C+\angle D={{360}^{o}} $

$ \angle A+\angle C={{360}^{o}}-{{160}^{o}} $

$ \angle A+\angle C={{200}^{o}} $

Hence, this is the answer.