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

Area of a triangle with sides √6 and 6-√6 is equal to

  1. √6 × 6-√6= 6√6 -√6

  2. √6 × 6-√6 = 6√6 - 6

  3. √6 ×(6-√6)= 6√6 - 6

  4. √6 ×(6-√6)= 6√6 -√6
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The area is calculated by multiplying the base and height (or sides in this context), which is sqrt(6) * (6 - sqrt(6)). Distributing the sqrt(6) gives sqrt(6)*6 - sqrt(6)*sqrt(6), which simplifies to 6*sqrt(6) - 6.

Multiple choice maths properties of parallel lines and their transversal introduction to shapes similarity of triangles introduction to similar triangles

Ratio of two corresponding sides of two similar triangles is $4:9$. Then ratio of their area is ___.

  1. $\dfrac{16} {81}$
  2. $\dfrac{34} {81}$
  3. $\dfrac{81} {16}$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Ratio of areas of two similar triangles is equal to the squares of the ratio of their sides.

Ratio of sides $=\dfrac{4}{9}$
Ratio of areas $=\left( \dfrac { 4 }{ 9 }  \right) ^{ 2 }=\dfrac { 16 }{ 81 } $

Multiple choice maths properties of parallel lines and their transversal introduction to shapes similarity of triangles introduction to similar triangles

$\triangle PQR \sim \triangle XYZ, \dfrac{XY}{PQ}=\dfrac{3}{2}$ then $\dfrac{Area\ of\ \triangle PQR}{Area\ of\ \triangle XYZ}=$____.

  1. $\dfrac{9}{4}$
  2. $\dfrac{4}{9}$
  3. $\dfrac{3}{2}$
  4. $\dfrac{2}{3}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\dfrac{{XY}}{{PQ}} = \dfrac{3}{2}$


$ \Rightarrow \dfrac{{PQ}}{{XY}} = \dfrac{2}{3}$


Now,  $\dfrac{{Area{\rm{ of  }}\Delta {\rm{PQR}}}}{{Area{\rm{ of  }}\Delta {\rm{XYZ}}}} = {\left( {\dfrac{2}{3}} \right)^2} = \dfrac{4}{9}$

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

The perimeter of two similar triangles are $24$ cm and $16$ cm, respectively. If one side of the first triangle is $10$ cm, then the corresponding side of the second triangle is

  1. $9$ cm
  2. $\dfrac{20}3$ cm
  3. $\dfrac{16}3$ cm
  4. $5$ cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
In similar triangles, ratio of the sides is equal to the ratio of the perimeters.
Thus, $\dfrac{p _1}{p _2} = \dfrac{s _1}{s _2}$
$\dfrac{24}{16} = \dfrac{10}{s _2}$
$s _2 = \dfrac{20}{3}$
Thus, side of the other triangle is $\dfrac{20}{3}$ cm.
Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

The area of two similar triangles $\displaystyle \Delta ABC$ and $\displaystyle \Delta DEF$ are 144 $\displaystyle cm^{2}$ and 81 $\displaystyle cm^{2}$ respectively If the longest side of larger $\displaystyle \Delta ABC$ be 36 cm then the longest side of the smaller triangle $\displaystyle \Delta DEF$ is

  1. 20 cm

  2. 26 cm

  3. 27 cm

  4. 30 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In similar triangle ABC & DBF
$\frac{AB}{De}=\frac{BC}{EF}=\frac{AC}{DF}=\frac{ratio ofArea of triangleABC}{ratio ofArea of triangleDEF}$ 
THEN $\frac{9}{12}=\frac{x}{36}$  (where x is longest side of the smaller triangle )
So x=27 cm

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

The perimeters of two similar triangles are $25\;cm$ and $15\;cm$ respectively. If one side of first triangle is $9\;cm$, then the corresponding side of the other triangle is

  1. $6.2\;cm$
  2. $3.4\;cm$
  3. $5.4\;cm$
  4. $8.4\;cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$The\quad perimeter\quad of\quad triangle\quad is\quad 25cm\quad and\quad 15cm.\ The\quad ratio\quad of\quad Perimeter\quad of\quad triangle\quad is\quad 25:15=5:3\ The\quad first\quad side\quad is\quad 9cm\quad ,let\quad the\quad other\quad side=x\ Hence,\quad \dfrac { 9 }{ x } =\dfrac { 5 }{ 3\  } \ \Rightarrow x=\dfrac { 3\times 9 }{ 5 } =\dfrac { 27 }{ 5 } =5.4\quad cm$

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

Sides of two similar triangles are in the ratio of $5 : 11$ then ratio of their areas is 

  1. $25 : 11$
  2. $25 : 121$
  3. $125 : 121$
  4. $121 : 25$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since, ratio of area of two similar traingles = ratio of square of corresponding sides

 ratio of sides = 5 : 11
$\therefore$ ratio of their areas = $(5)^2 : (11)^2 = 25 : 121$

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

Sides of two similar triangles are in the ratio of $4 : 9$ then area of these triangles are in the ratio

  1. $2 : 3$
  2. $4 : 9$
  3. $81 : 16$
  4. $16 : 81$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle \because \Delta ABC\sim \Delta DEF$
$\displaystyle \therefore \dfrac{ar\Delta ABC}{ar\Delta DEF}=\dfrac{\left ( 4 \right )^{2}}{\left ( 9 \right )^{2}}=\dfrac{16}{81}=16:81.$

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

In two similar triangles ABC and PQR, if their corresponding altitudes AD and Ps are in the ratio 4:9, find the ratio of the areas of $\triangle ABC$ and $\triangle PQR$.

  1. $16:81$
  2. $9:16$
  3. $81:16$
  4. $16:9$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Since the areas of two similar triangles are in the ratio of the squares of the corresponding altitudes.

$\therefore $ $\dfrac { Area(\triangle ABC) }{ Area(\triangle PQR) } =\dfrac { { AD }^{ 2 } }{ { PS }^{ 2 } } $

$\Rightarrow $ $\dfrac { Area(\triangle ABC) }{ Area(\triangle PQR) } ={ \left( \dfrac { 4 }{ 9 }  \right)  }^{ 2 }=\dfrac { 16 }{ 81 } $              [$\because AD:PS=4:9$]

$\Rightarrow $ $\dfrac { Area(\triangle ABC) }{ Area(\triangle PQR) }$ = $\dfrac{16}{81}$
Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

If $\triangle ABC$ is similar to $\triangle DEF$ such that BC=3 cm, EF=4 cm and area of $\triangle ABC=54 {cm}^{2}$. Determine the area of $\triangle DEF$.

  1. $40\ cm^2$
  2. $59\ cm^2$
  3. $69\ cm^2$
  4. $96\ cm^2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Since the ratio of areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides.

$\therefore $ $\dfrac { Area(\triangle ABC) }{ Area(\triangle DEF) } =\dfrac { { BC }^{ 2 } }{ { EF }^{ 2 } } $

$\Rightarrow $ $\dfrac { 54 }{ Area(\triangle DEF) } =\dfrac { { 3 }^{ 2 } }{ { 4 }^{ 2 } } $

$\Rightarrow $ $Area(\triangle DEF)=\dfrac { 54\times 16 }{ 9 } =96{ cm }^{ 2 }$
Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

If $A={30}^{\circ},\,a=100,\,c=100\sqrt{2}$, find the number of triangles that can be formed.

  1. $1$
  2. $2$
  3. $3 $
  4. $4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Here $a, c$ and $A$ are given, $\therefore$ we will have to examine whether two triangle are possible or not. For two triangles
$(i)\,a>c\sin{A}$ and $(ii)a<c$
$\Rightarrow 100>100\sqrt{2}\sin{{30}^{\circ}}$
$\Rightarrow 100>100\sqrt{2}\times\dfrac{1}{2}$
$\Rightarrow 100>50\sqrt{2}$
and $a<c$
i.e., $100<100\sqrt{2}$
$\Rightarrow $ Two triangles can be formed.
Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

In $\triangle ABC \sim \triangle DEF$ such that $AB = 1.2\ cm$ and $DE = 1.4\ cm$. Find the ratio of areas of $\triangle ABC$ and $\triangle DEF$.

  1. $36 : 50$
  2. $49 : 50$
  3. $36 : 49$
  4. $1:2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We know that area of two similar triangle is equal to the ratio of the squares of any two corresponding sides
$\dfrac {ar(\triangle ABC)}{ar (\triangle DEF)} = \dfrac {AB^{2}}{DE^{2}} = \dfrac {(1.2)^{2}}{(1.4)^{2}} = \dfrac {36}{49}$

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

The perimeter of two similar triangle are $30\ cm$ and $20\ cm$. If one side of first triangle is $12\ cm$ determine the corresponding side of second triangle.

  1. $8\ cm$
  2. $4\ cm$
  3. $3\ cm$
  4. $16\ cm$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the two similar triangles be $\triangle ABC$ and $\triangle DEF$

$\therefore \dfrac {AB}{DE} = \dfrac {BC}{EF} = \dfrac {AC}{DF} = \dfrac {P _{1}}{P _{2}}$

$\Rightarrow \dfrac {AB}{DE} = \dfrac {P _{1}}{P _{2}}$

$\Rightarrow \dfrac {12}{DE} = \dfrac {30}{20}$

$\Rightarrow DE = 8\ cm$

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

If a triangle with side lengths as $5, 12$, and $15$ cm is similar to a triangle which has longer side length as $24$ cm, then the perimeter of the other triangle is:

  1. $38.4$
  2. $44$
  3. $51.2$
  4. $58$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The longer side of the bigger triangle is $24$ cm.

The longer side of the smaller triangle is $15$ cm.
They are in ratio $24:15 = \cfrac{24}{15} = 1.6$
Thus, their perimeters also would be in the ratio $1.6$
The perimeter of the smaller triangle is $5 + 12 + 15 = 32$ cm
Implies the perimeter of the bigger triangle would be $32 \times 1.6 = 51.2$ cm