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 similarity relation between perimeters of similar shapes basic proportionality theorem and its converse basic proportionality theorem

In a $\Delta ABC$, let $M$ be the mid-point of segment $AB$ and let $D$ be the foot of the bisector of $\angle C$. Then the ratio $\dfrac{Area\Delta CDM}{Area \Delta ABC}$ is $\left(A>B\right)$

  1. $\dfracc{1}{4}\dfrac{a-b}{a+b}$
  2. $\dfracc{1}{2}\dfrac{a-b}{a+b}$
  3. $\dfracc{1}{2}\tan\dfrac{A-B}{2}\cot\dfrac{A+B}{2}$
  4. $\dfracc{1}{4}\cot\dfrac{A-B}{2}\tan\dfrac{A+B}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths similarity relation between perimeters of similar shapes basic proportionality theorem and its converse basic proportionality theorem

The sides of a triangle are $3x+4y,\,4x+3y$ and $5x+5y$ units, where $x,y>0$.The triangle is ______________.

  1. right angled

  2. equilateral

  3. obtuse angled

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let $a=3x+4y,\,b=4x+3y$ and $c=5x+5y$ be the largest side
$\Rightarrow \cos{C}=\dfrac{{a}^{2}+{b}^{2}-{c}^{2}}{2ab}$
$=\dfrac{{\left(3x+4y\right)}^{2}+{\left(4x+3y\right)}^{2}-{\left(5x+5y\right)}^{2}}{2\left(3x+4y\right)\left(4x+3y\right)}$
$\Rightarrow \cos{C}=\dfrac{9{x}^{2}+16{y}^{2}+24xy+16{x}^{2}+9{y}^{2}+24xy-25{x}^{2}-25{y}^{2}-50xy}{2\left(3x+4y\right)\left(4x+3y\right)}<0,\,\,\,x,y>0$
$\Rightarrow \cos{C}=\dfrac{-2xy}{2\left(3x+4y\right)\left(4x+3y\right)}<0,\,\,x,y>0$
$\Rightarrow \theta>{90}^{\circ}$
$\therefore,\, $ the triangle is obtuse angled.
Multiple choice maths similarity relation between perimeters of similar shapes basic proportionality theorem and its converse basic proportionality theorem

D and E are respectively the points on the sides AB and AC of a $\displaystyle \Delta ABC$ such that $AB = 12 cm$, $AD = 8 cm$, $AE = 12 cm$ and $AC = 18 cm$, then

  1. DE $\parallel$ BD is true
  2. DE $\parallel$ BC is true
  3. AD $\parallel$ BD is true
  4. AD $\parallel$ CD is true
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We have,
AB = 12 cm, AC = 18 cm, AD = 8 cm and AE = 12 cm.
$\displaystyle \therefore \quad BD=AB-AD=\left( 12-8 \right) cm=4cm$
$\displaystyle CE=AC-AE=\left( 18-12 \right) cm=6cm$
Now, $\displaystyle \frac { AD }{ BD } =\frac { 8 }{ 4 } =\frac { 2 }{ 1 } $
And, $\displaystyle \frac { AE }{ CE } =\frac { 12 }{ 6 } =\frac { 2 }{ 1 } $
$\displaystyle \Rightarrow \quad \frac { AD }{ BD } =\frac { AE }{ CE } $
Thus, DE divides sides AB and AC of $\displaystyle \Delta ABC$ in the same ratio. Therefore, by the converse of basic proportionality theorem, we have
$\displaystyle DE\parallel BC$.

Multiple choice maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

Let  $z _ { 1 } , z _ { 2 }$  and  $z _ { 3 }$  represent the vertices  $A, B$  and  $C$  of the triangle  $A B C$  in the argand that  $\left| z _ { 1 } \right| = \left| z _ { 2 } \right| = \left| z _ { 3 } \right| = 5,$  then  $z _ { 1 } \sin 2 A + z _ { 2 } \sin 2 B + z _ { 3 } \sin 2 C = 0.$

  1. True

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

This is a known identity in complex geometry for points on a circle centered at the origin. The sum of the vectors weighted by the sine of the angles relates to the geometry of the triangle inscribed in the circle.

Multiple choice maths trigonometry trigonometric ratios of acute angles compound angles, multiple angles, sub multiple angles and transformation formulae trigonometric identities

One angle of a triangle is $\displaystyle \frac{2x}{3}$ grades another is $\displaystyle \frac{3x}{2}$ degrees, whilst the third is $\displaystyle \frac{2\pi x}{75}$ radians ; express them all in degrees.

  1. ${ 55 }^{ o },\quad { 28 }^{ o }\quad \& \quad { 97 }^{ o }\\$
  2. ${ 65 }^{ o },\quad { 22 }^{ o }\quad \& \quad { 93 }^{ o }\\$
  3. $\\{ 60 }^{ o },\quad { 24 }^{ o }\quad \& \quad { 96 }^{ o }\\$
  4. ${ 70 }^{ o },\quad { 15 }^{ o }\quad \& \quad { 95 }^{ o }\\$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Convert all angles to degrees: (2x/3) grades = (2x/3) * (9/10) = 0.6x degrees. (3x/2) degrees = 1.5x degrees. (2*pi*x/75) radians = (2*pi*x/75) * (180/pi) = 4.8x degrees. Sum = 0.6x + 1.5x + 4.8x = 6.9x = 180. x = 180/6.9 = 26.08. Checking the options, C gives 60, 24, 96 which sum to 180.

Multiple choice maths enlargement and scale drawing dilation enlargement similarity as a size transformation mapping mapping space around us bearing and drawings

A $30-60-90$ degree triangle is scaled $1.5$ times. The new angles of the triangle are:

  1. $45-45-90$
  2. $30-60-90$
  3. $37-53-90$
  4. $60-60-60$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

When the triangle is scaled $1.5$ times then length of each side become $1.5$ times but the angle remains the same.

So the new angles are $30-60-90$
Option $B$ is correct.

Multiple choice maths enlargement and scale drawing dilation enlargement similarity as a size transformation mapping mapping space around us bearing and drawings

The ratio of the lengths of the corresponding sides of $2$ similar right angled triangles is $2:5$. If the length of the hypotenuse of the smaller triangle is $5$ inches, find the length of the hypotenuse of the larger triangle (in inches):

  1. 2

  2. 2.5

  3. 7

  4. 10

  5. 12.5
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Ratio of the length of the sides of the two triangle $=2:5$

If hypotenuse  of small triangle $=5$ inches
Let the hypotenuse of  larger triangle $=x$
$\therefore \dfrac{5}{x}=\dfrac{2}{5}$
$\therefore  x=\dfrac{25}{2}=12.5$  inches

Multiple choice mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

The degree measure of 1 radian (taking $\pi =\dfrac { 22 }{ 7 }$ ) is

  1. $55^o{ 61 }^{ ' }{ 22 }^{ " }$ (approx.)
  2. $57^o{ 16 }^{ ' }{ 22 }^{ " }$ (approx.)
  3. $57^o{ 22 }^{ ' }{ 16 }^{ " }$ (approx.)
  4. $57^o{ 22 }^{ ' }{ 22 }^{ " }$ (approx.)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\pi\ radians = 180^{\circ}$

$1\ radian=\frac { 180 }{ \pi  } = \frac { 180 }{ \frac { 22 }{ 7 }  } $
$1\ radian=57.272727$
The integer part constitutes the degree part. The mantissa is converted to minutes by multiplying with ${60}'$
Minutes = $0.272727*{60}' = {16.3636}'$
The integer part constitutes the minutes. The mantissa is converted to seconds by multiplying with ${60}''$
Seconds = $0.3636*{60}''\approx {22}''$
Hence, the degree measure of 1 radian is $57^{\circ}{16}'{22}''$

Multiple choice maths similarity relation between perimeters of similar shapes basic proportionality theorem and its converse basic proportionality theorem

In an isosceles $\Delta A B C$ the base $A B$ is produced both the ways to $P$ and $Q$ such that $A P \times BO = A C ^ { 2 }$ then $\Delta A P C \sim \Delta B C Q$

  1. True

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

Given the geometric properties of the isosceles triangle and the condition AP * BQ = AC^2, the triangles APC and BCQ satisfy the criteria for similarity (SAS similarity).

Multiple choice trigonometric equations trigonometric functions trigonometry maths

$A, B, C$ are three angles such that $\tan  A+\tan  B+\tan  C=\tan  A  \tan  B  \tan  C.$ Which of the following statements is always correct ?

  1. $ABC$ is a triangle, i.e. $A+B+C=\pi $
  2. $A=B=C. i.e., $ $ABC$ is an equilateral triangle
  3. $A+B=C, $ i.e., $ABC$ is a right- angled triangle
  4. $A+B=\pi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

(A) $tan\left [ (A+B)+C \right ]$
$=\frac{tan (A+B)+tan C}{1-tan (A+B)  tan  C}=\frac{\frac{tan  A+tan   B}{1-tan  A   tan  B}+tan  C}{1-\frac{tan  A+tan  B}{1-tan   A   tan  B}.  tan  C}$
$=\frac{tan  A+tan  B+tan  C-tan  A   tan  B   tan  C}{Denominator}$
$=0$
$\left [ since,  tan  A+tan  B+tan  C =tan  A   tan  B   tan  C \right ]$
$\therefore A+B+C=\pi $ i.e.,  A, B, C is a triangle

Multiple choice trigonometric equations trigonometric functions trigonometry maths

The cosine of the obtuse angle formed by the medians from the vertices of the acute angles of an isosceles right angled triangle is

  1. $- 2 / 3$
  2. $- 4 / 5$
  3. $- 3 / 5$
  4. $- 3 / 4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In an isosceles right triangle, placing vertices at (0,0), (a,0), and (0,a), the medians from the acute angles are calculated. The cosine of the angle between them is -2/3.

Multiple choice trigonometric equations trigonometric functions trigonometry maths

In an isosceles $\triangle ABC$, if the altitudes intersect on the inscribed circle then cosine of the vertical angle $'A'$ is :

  1. $\cfrac{1}{9}$
  2. $\cfrac{1}{3}$
  3. $\cfrac{2}{3}$

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

If the altitudes of an isosceles triangle intersect on the inscribed circle, the geometry dictates that cos(A) = 1/9.

Multiple choice polyhedrons visualising solid shapes maths

O ABC is a tetrahedron such that OA$=$OB$=$CO$=$K and $\angle AOB=\angle BOC =\angle COA =\theta$.

  1. $\left[\dfrac{\pi}{3}, \dfrac{2\pi}{3}\right]$
  2. $\left[0, \dfrac{2\pi}{3}\right]$
  3. $\left[\dfrac{\pi}{4}, \dfrac{\pi}{3}\right]$
  4. $\left[0, \dfrac{\pi}{2}\right]$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a tetrahedron OABC with OA=OB=OC=K and equal face angles theta at O, the dihedral angles and face angles are constrained by the geometry of the solid. The range of the face angle theta for such a tetrahedron is [pi/3, 2pi/3].

Multiple choice business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

The triangle formed by the lines whose combined equation is  $\displaystyle (y^{2}-4xy-x^{2}) ( x+y-1  )=0$ is

  1. equilateral.

  2. right angled.

  3. isosceles.

  4. obtuse angled.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given equation of the pair of straight lines is  $x^{2}-4xy-y^{2}=0$

Since, coefficient of $x^2$ + coefficient of $y^2$ $= 0$ 

The two lines are perpendicular.

Therefore, triangle formed is right angled.