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 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 $k$ 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

Mark the triplet that can be the lengths of the sides of a triangle.

  1. $2, 3, 5$
  2. $1, 4, 2$
  3. $7, 4, 4$
  4. $5, 6, 12$
  5. $9, 20, 8$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

for triplet to be the length of triangle, it must satisfy triangle property.

sum of any two sides must be greater than third side
So, $7,4,4$ is correct answer.
If we consider remaining options, observe that sum of two sides is less than third side, so they cannot form triangle.

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

$D$ is a point on the side $BC$ of a $\Delta$ $ABC$, such that $AD$ bisects $\angle $ $BAC$. Then:

  1. $BA = CD$
  2. $BA > BD$
  3. $BD > BA$
  4. $CD > CA$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The definition of the angle bisector of a triangle is a line segment that bisects one of the vertex angles of a triangle. In general, an angle bisector is equidistant from the sides of the angle when measured along a segment perpendicular to the sides of the angle 

Hence, by definition,$BD+DC=BC$
Therefore, $BA>BD$

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

Let a,b,c be the sides of a triangle. No two of them are equal and $\lambda  \in R.$ If the roots of the equation 

  1. $\lambda < \frac{4}{3}$
  2. $\lambda < \frac{5}{3}$
  3. $\lambda \in \left( {\frac{1}{3},\frac{5}{3}} \right)$
  4. $\lambda \in \left( {\frac{4}{3},\frac{5}{3}} \right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
The question must be Let $a,b,c$ be the sides of a triangle.No two of them are equal and $\lambda\in R$. If the roots of the eqn ${x}^{2}+2\left(a+b+c\right)x + 3\lambda\left(ab+bc+ca\right) = 0$ are real, then $\lambda\in\,R$

Given, roots of equation ${x}^{2}+2\left(a+b+c\right)x + 3\lambda\left(ab+bc+ca\right) = 0$ are real,
So,$D\ge\,0$

$\Rightarrow\,{\left[2\left(a+b+c\right)\right]}^{2}-4\times 1\times 3\lambda\left(ab+bc+ca\right)\ge 0$

$\Rightarrow\,4{\left(a+b+c\right)}^{2}-12\lambda\left(ab+bc+ca\right)\ge 0$

$\Rightarrow\,4{\left(a+b+c\right)}^{2}\ge\,12\lambda\left(ab+bc+ca\right)$

$\Rightarrow\,\lambda\le\dfrac{4{\left(a+b+c\right)}^{2}}{12\lambda\left(ab+bc+ca\right)}$

$\Rightarrow\,\lambda\le\,\dfrac{4\left({a}^{2}+{b}^{2}+{c}^{2}\right)}{12\lambda\left(ab+bc+ca\right)}+\dfrac{8\lambda\left(ab+bc+ca\right)}{12\lambda\left(ab+bc+ca\right)}$

$\Rightarrow\,\lambda\le\,\dfrac{4\left({a}^{2}+{b}^{2}+{c}^{2}\right)}{12\lambda\left(ab+bc+ca\right)}+\dfrac{8}{12}$

$\Rightarrow\,\lambda\le\,\dfrac{4\left({a}^{2}+{b}^{2}+{c}^{2}\right)}{12\lambda\left(ab+bc+ca\right)}+\dfrac{2}{3}$      .......$(1)$

Now, we know that,

$\left|a-b\right|<c\Rightarrow\,{a}^{2}+{b}^{2}-2ab<{c}^{2}$

$\left|b-c\right|<a\Rightarrow\,{b}^{2}+{c}^{2}-2bc<{a}^{2}$

$\left|c-a\right|<c\Rightarrow\,{c}^{2}+{a}^{2}-2ac<{b}^{2}$

On adding,

${a}^{2}+{b}^{2}-2ab+{b}^{2}+{c}^{2}-2bc+{c}^{2}+{a}^{2}-2ac<{a}^{2}+{b}^{2}+{c}^{2}$

${a}^{2}+{b}^{2}+{c}^{2}<2ab+2bc+2ca$

$\Rightarrow\,dfrac{{a}^{2}+{b}^{2}+{c}^{2}}{ab+bc+ca}<2$

So, eqn$(1)$ becomes,

$\lambda<\dfrac{2}{3}+\dfrac{2}{3}$

$\therefore\,\lambda<\dfrac{4}{3}$
Multiple choice physics mathematical methods multiplication of vectors products of vectors scalar product

If $\overrightarrow{A}=4\widehat{i}+6\widehat{j} $  and  $\overrightarrow{B}=2\widehat{i}+3\widehat{j}$ .Then :

  1. $\overrightarrow{A}.\overrightarrow{B} =29$
  2. $\overrightarrow{A}\times \overrightarrow{B}=0$
  3. $\dfrac{|\overrightarrow{A}|}{|\overrightarrow{B}|}=\dfrac{2}{1} $
  4. angles between $ \overrightarrow{A}$ and $\overrightarrow{B} $ is $ 30^{\circ}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
(A) $\vec A.\vec B=(4\hat i+6\hat j).(2\hat i+3\hat j)=8+18=26$

(B) $\overrightarrow{A} \times \overrightarrow{B}= \begin{vmatrix} \widehat{i}& \widehat{j} &\widehat{k}\\  4& 6  & 0\\ 2 & 3 & 0\end{vmatrix} = \widehat{i}(0-0)-\widehat{j}(0-0)+\widehat{k}(12-12)=0 $

(C) $|\vec A|=\sqrt{4^2+6^2}=7.21$
     $|\vec B|=\sqrt{2^2+3^2}=3.60$
     $\dfrac{|\vec A|}{|\vec B|}=2$

(D) Angle between $\vec A$ and $\vec B$ $= cos ^{-1}\dfrac{\vec A.\vec B}{|\vec A|.|\vec B|}=cos^{-1}1=0^0$
Multiple choice maths construction of parallel lines and triangles triangle inequality inequalities in triangle inequalities in triangles

The points $\left( 0,\dfrac { 8 }{ 3 }  \right),(1,3)$ and $(82,30)$ are the vertices of:

  1. an equilateral triangle

  2. an isosceles triangle

  3. a right angled triangle

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

According to the problem :

$AB^2=(0-1)^2+(\dfrac{8}{3}-3)^2$
$=1+\dfrac{1}{9}=\dfrac{10}{9}=1.11$

Similarly,
$BC^2=(82-1)^2+(30-3)^2=7290$
and
$AC^2=(82-0)^2+(30-\dfrac{8}{3})^2=7471.11$

Therefore,
$AB^2+BC^2<AC^2$

Hence the answer is acute-angled triangle.

Multiple choice maths congruency of triangles triangle inequality inequalities in triangle inequalities in triangles

The triangle inequality theorem states that 

  1. The sum of the lengths of the $2$ sides of a triangle is equal than the third side of the triangle
  2. The sum of the lengths of the $2$ sides of a triangle is less than the third side of the triangle
  3. The sum of the lengths of the $2$ sides of a triangle is more than the third side of the triangle

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

The triangle inequality theorem states that the sum of the lengths of the $2$ sides of a triangle is greater than the third side of the triangle.

Multiple choice maths congruency of triangles triangle inequality inequalities in triangle inequalities in triangles

State the following statement is True or False
The triangle inequality theorem states that the sum of the lengths of the $2$ sides of a triangle is equal than the third side of the triangle

  1. True

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

The triangle inequality theorem states that the sum of the lengths of the $2$ sides of a triangle is greater than the third side of the triangle.