Mathematics · Quantitative Aptitude

Triangle Properties

234 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

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 surface area and volume of cube and cuboid finding out the diagonal of cube and cuboid length of the diagonal of cube diagonal of cube and cuboid

What is the value of x, if (7, x, x - 2) is a Pythagorean triple?

  1. -11.25

  2. 26.5

  3. -16.5

  4. 16.5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Applying the Pythagorean triples rule as $a^{2}+b^{2}= c^{2}$
$7^{2}+x^{2}= (x - 2)^{2}$
49 + $x^{2}$ = $x^{2}$+ 4 -4x
Both the side $x^{2}$ will get cancelled,
45 = -4x
x = -11.25

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 theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

If every side of a triangle is doubled, then the area of the new triangle is 'K' times the area of the old one. The value of K is

  1. 2

  2. 3
  3. $\sqrt 2$

  4. 4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Let the area of the triangle be $x$.

We know that the area of the triangle
$=\dfrac{1}{2}\times Height \times Base$
$x=\dfrac{1}{2}\times Height \times Base$              $........ (1)$

According to the question,
$Kx=\dfrac{1}{2}\times 2 \times Height \times 2 \times Base$
$Kx=4\times x$
$K=4$

Hence, this is the answer.