Mathematics · Quantitative Aptitude

Triangle Properties

243 Questions

Triangle properties encompass the rules governing the sides, angles, and area of different types of triangles. Key areas include the Pythagorean theorem, similar triangles, and centroid calculations. These concepts form a foundational part of geometry in various competitive examinations.

Similar trianglesArea and perimeterPythagorean theoremTriangle inequalityEquilateral properties

Triangle Properties Questions

Multiple choice mathematics and statistics inverse of a matrix and linear equations application of determinants area of triangle and collinearity of three points applications of determinants

The vertices of the triangle $ABC$ are $(2, 1, 1), (3, 1, 2), (-4, 0, 1)$. The area of triangle is

  1. $\displaystyle \frac{3\sqrt{38}}{2}$
  2. $\sqrt{38}$
  3. $\displaystyle \frac{\sqrt{38}}{2}$
  4. $4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The vertices of the triangle $ABC$ are $(2,1,1),(3,1,2),(-4,0,1)$
$\overrightarrow { AB } =i+k$ and $\overrightarrow { AC } =-6i-j$
now, $\displaystyle \triangle =\frac { \left| \overrightarrow { AB } \times \overrightarrow { AC }  \right|  }{ 2 } =\frac { \left| \left( i+k \right) \times \left( -6i-j \right)  \right|  }{ 2 } =\frac { \left| i-6j-k \right|  }{ 2 } $
Therefore, $\triangle =\dfrac { \sqrt { 38 }  }{ 2 } $

Ans: C

Multiple choice mathematics and statistics inverse of a matrix and linear equations application of determinants area of triangle and collinearity of three points applications of determinants

What is the area of the triangle formed by the points $(a,c+a), (a,c)$ and $(-a,c-a)$?

  1. $\displaystyle- a^{2}$
  2. $\displaystyle \frac{1}{a^{2}}$
  3. $\displaystyle a^{2}+a$

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

$\left( a,c+a \right)  \left( a,c \right)  \left( -a,c-a \right) $

$\triangle \begin{vmatrix} 1 & 1 & 1 \ a & a & -a \ c+a & \quad c & \quad c-a \end{vmatrix}$
$ac-{ a }^{ 2 }+ac-(ac-{ a }^{ 2 }+ac+{ a }^{ 2 })+ac-ac-{ a }^{ 2 }$
$-2{ a }^{ 2 }-2ac+2ac$
$-2{ a }^{ 2 }$
Area $=\cfrac { 1 }{ 2 } \left[ \triangle  \right] =\cfrac { 1 }{ 2 } \left( -{ a }^{ 2 } \right) $
$=-{ a }^{ 2 }$

Multiple choice mathematics and statistics inverse of a matrix and linear equations application of determinants area of triangle and collinearity of three points applications of determinants

What is the area of the triangle formed by the points $(a,c+a), \displaystyle \left ( a^{2},c^{2} \right )$ and $(-a, c-a)$?

  1. $1$
  2. $\displaystyle a^{2}$
  3. $\displaystyle \sqrt{a^{2}+c^{2}}$

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

$(a,c+a)\quad ({ a }^{ 2 },{ c }^{ 2 })\quad (-a,\quad c-a)$

$\triangle =\begin{vmatrix} 1 & 1 & 1 \ a & { a }^{ 2 } & -a \ c+a & { \quad c }^{ 2 } & \quad c-a \end{vmatrix}$
${ a }^{ 2 }c-{ a }^{ 3 }+a{ c }^{ 2 }-ac+{ a }^{ 2 }-ac-{ a }^{ 2 }+a{ c }^{ 2 }+{ a }^{ 2 }c-{ a }^{ 3 }$
$2a{ c }^{ 2 }-2{ a }^{ 3 }-2ac$
Area $=\cfrac { 1 }{ 2 } \triangle $
$=a{ c }^{ 2 }-{ a }^{ 3 }-ac$
OPTION-D

Multiple choice mathematics and statistics inverse of a matrix and linear equations application of determinants area of triangle and collinearity of three points applications of determinants

What is the area of the triangle formed by the points $(a,b+c), (b,c+a)$ and $(c,a+b)$?

  1. $1$
  2. $-1$
  3. $0$
  4. $\displaystyle \frac{1}{2}\left ( abc \right )^{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$\left( a,b+c \right) ;\left( b,c+a \right) ;\left( c,a+b \right) $
$\triangle =\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ b+c & c+a & a+b \end{vmatrix}$
$ab+{ b }^{ 2 }-{ c }^{ 2 }-ac-{ a }^{ 2 }-ab+bc+{ c }^{ 2 }+ac+{ c }^{ 2 }-{ a }^{ 2 }-bc$
$=0$
Area of triangle$=\cfrac { 1 }{ 2 } \left| \triangle  \right| =0$
Option C
Multiple choice mathematics and statistics inverse of a matrix and linear equations application of determinants area of triangle and collinearity of three points applications of determinants

The area of a triangle whose vertices are (-2,-2), (-1,-3) and (p,0) is 3 sq.units what is the value of p?

  1. -2

  2. 2

  3. 3

  4. -3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the vertices of the triangle A(-2,-2),B(-1,-3) and C(P,0) then

$Area  of   \triangle ABC=\frac{1}{2}\left[x _1(y _2-y _3)+x _2(y _3-y _1)+x _3(y _1-y _2)\right]$
Area =3 sq. unit
Here$x _1=-2,y _1=-2$
        $x _2=-1,y _2=-3$
         $x _3=p,y _3=0$
$\Rightarrow 3=\frac{1}{2}\left[(-2(-3-0)+-1(0-(-2))+p(-2-(-3)\right]$
$\Rightarrow 3=\frac{1}{2}[6-2+p]$
$\Rightarrow 6=4+p$
$\Rightarrow -p=4-6$
$\Rightarrow p=2$

Multiple choice mathematics and statistics inverse of a matrix and linear equations application of determinants area of triangle and collinearity of three points applications of determinants

The area of a triangle, whose vertices are $(3, 2), (5, 2)$ and the point of intersection of the lines $x = a$ and $y = 5$, is $3$ square units. What is the value of $a$?

  1. $2$
  2. $3$
  3. $4$
  4. $5$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$A(3,2) B(5,2)$ intersection of $x=a,y=5$   $(a,5)$
$\cfrac { 1 }{ 2 } \left| \triangle  \right| =3$
$\triangle =6$
$=\begin{vmatrix} 1 & 1 & 1 \\ 3 & 5 & a \\ 2 & 2 & 5 \end{vmatrix}$
$=25-2a-15+2a+6-10$
$=6$
For any value of a area $=3sq.units$
Multiple choice maths similarity relation between perimeters of similar shapes basic proportionality theorem and its converse basic proportionality theorem

The areas of two similar triangle are $18\ cm^{2}$ and $32\ cm^{2}$ respectively. What is the ratio of their corresponding sides?

  1. $3:4$
  2. $4:3$
  3. $9:16$
  4. $16:9$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The ratio of areas of similar triangles is the square of the ratio of their corresponding sides. sqrt(18/32) = sqrt(9/16) = 3/4.

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

A triangle ABC has been enlarged by scale factor m= 2.5 to the triangle A' B' C'. Calculate the length of C' A' if CA=4 cm.

  1. 10

  2. 8

  3. 6

  4. 12
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\triangle ABC$ is enlarged to $\triangle A'B'C'$,
Thus, $\dfrac{C'A'}{CA} = 2.5$
Therefore, $\dfrac{C'A'}{4} = 2.5$
$\Rightarrow C'A' = 4 \times 2.5$
$\Rightarrow C'A' = 10$ cm 

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

A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find OA, if OA'= 6 cm.

  1. 2 cm

  2. 3 cm

  3. 4 cm

  4. none of the above

Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\triangle ABC$ is enlarged to $\triangle A'B'C'$,
Thus, $\dfrac{OA'}{OA} = 3$
$\Rightarrow \dfrac{6}{OA} = 3$
$\Rightarrow OA = \dfrac{6}{3}$
$\Rightarrow OA = 2 $ cm

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

A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find BC. if B'C'= 15 cm.

  1. 6 cm

  2. 5 cm

  3. 7 cm

  4. none of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\triangle ABC$ is enlarged to $\triangle A'B'C'$,
Thus, $\dfrac{B'C'}{BC} = 3$
$\Rightarrow \dfrac{B'C'}{BC} = 3$
$\Rightarrow BC = \dfrac{15}{3}$
$\Rightarrow BC = 5$ cm 

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

A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find A'B', if AB = 4cm.

  1. 12 cm

  2. 14 cm

  3. 22 cm

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

$\triangle ABC$ is enlarged to $\triangle A'B'C'$,
Thus, $\dfrac{A'B'}{AB} = 3$
$\Rightarrow \dfrac{A'B'}{4} = 3$
$\rightarrow A'B' = 4 \times 3$
$\Rightarrow A'B' = 12 $ cm 

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

A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find OC', if  OC=21 cm.

  1. 63 cm

  2. 53 cm

  3. 43 cm

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

$\triangle ABC$ is enlarged to $\triangle A'B'C'$,
Thus, $\dfrac{OC'}{OC} = 3$
$\Rightarrow \dfrac{OC'}{21} = 3$
$\Rightarrow OC' = 21 \times 3$
$\Rightarrow OC' = 63 $ cm

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

Two mutually perpendicular straight lines are drawn from the origin to form an isosceles triangle with the straight line $\displaystyle x\cos \alpha +y\sin \alpha -p=0$. Then the area of this triangle is

  1. independent of $\displaystyle \alpha$

  2. independent of p
  3. independent of both $\displaystyle \alpha$ and p
  4. a function of $\displaystyle \alpha$ and p
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Two mutually perpendicular lines drawn from origin i.e X-axis and Y-axis 

$y=0$ and $x=0$
The area of triangle $=\dfrac{1}{2}\times x \times y$

Multiple choice maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

A regular triangular pyramid has an altitude of $9\ m$ and a volume of $187.06\ cu.\ m$. What is the base edge in meters?

  1. $12$
  2. $13$
  3. $14$
  4. $15$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given : Altitude$(height\quad (h))=9\ m$, volume $=187.06\ cu.\ m$

We know that, Volume $=\dfrac{1}{3}Bh$, where $B=x^2 \sin \theta$
$\implies 187.06=\dfrac{1}{3} \left(\dfrac{1}{2}x^2 (\sin \theta)\right) (9)$,  where $x$ is base edge
$\implies 187.06=\dfrac{1}{3} \left(\dfrac{1}{2}x^2 \sin60\right)9$
$\implies x^2=143.9988=144$
$\therefore\ x=12\ m$

Multiple choice maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

The frustum of a regular triangular pyramid has equilateral triangles for its bases. The lower and upper base edges are $9\ m$ and $3\ m$, respectively. If the volume is $118.2\ cu.\ m$, how far apart (m) are the base?

  1. $9$
  2. $8$
  3. $7$
  4. $10$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given : Volume $=118.2\ cu. m$

Upper base edge $=9\ m$, lower base edge $=3\ m$
We know that, 
Volume $=\dfrac{h}{3}(A _{1}+A _{2}+\sqrt{A _{1} A _{2}})$ ....... $(1)$, where $A _{1}, A _{2}$ are area of upper and lower bases.
$A _{1}=\dfrac{\sqrt{3}}{4}\times 9^2=35.074$
$A _{2}=\dfrac{\sqrt{3}}{4}\times 3^2=3.897$
From $(1)$ we get,
$118.2 = \dfrac{h}{3}(35.074+3.897+\sqrt{35.074\times 3.897})$
$\implies 118.2=\dfrac{h}{3}(38.971+11.6911)$
$\implies 118.2\times 3=h(50.6621)$
$\implies h=7\ m$
Hence, the bases are $7\ m$ far from each other.