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Multiple choice maths similarity areas of similar figures areas of similar triangles relations between the areas of triangles

A vertical stick of length $6m$ casts a shadow $4m$ long on the ground and at the same time a tower casts a shadow $28m$ long. Find the height of the tower.

  1. $42m$
  2. $48m$
  3. $62m$
  4. $52m$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The length of stick corresponds to the length of tower and shadow of stick corresponds to the shadow of tower.
Thus, $\dfrac{Length\ Stick}{Shadow\ Stick} = \dfrac{Length\ Tower}{Shadow\ Tower}$


$\dfrac{6}{4} = \dfrac{Length\ Tower}{28}$
$Length\ Tower = \dfrac{6 \times 28}{4}$
$Length\ Tower = 42$ m

Multiple choice maths similarity areas of similar figures areas of similar triangles relations between the areas of triangles

The corresponding sides of two similar triangles are in the ratio $2$ to $3$. If the area of the smaller triangle is $12$ the area of the larger is

  1. $24$
  2. $27$
  3. $18$
  4. $8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Area of similar triangles are in the ratio of square of the corresponding sides.
Hence, $\dfrac{\text{Area of smaller triangle}}{\text{Area of larger triangle}} = \frac{2^2}{3^2}$
$\Rightarrow \dfrac{12}{\text{Area of larger triangle}} = \dfrac{4}{9}$
$\text{Area of larger triangle}  = 27$

Multiple choice maths similarity areas of similar figures areas of similar triangles relations between the areas of triangles

If  in $\triangle ABC$  and $\triangle EDA,$ $\displaystyle BC\bot AB,AE\bot AB$ and $\displaystyle DE\bot AC$ then $\displaystyle DE.BC=AD.AB$ 

  1. True

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

In $\displaystyle \Delta ABC$ and $\displaystyle \Delta EDA$,
We have
$\displaystyle \angle ABC=\angle ADE$ [Each equal to $\displaystyle { 90 }^{ o }$]
$\displaystyle \angle ACB=\angle EAD$ [Alternate angles]
$\displaystyle \therefore $ By AA Similarity
$\displaystyle \Delta ABC\sim \Delta EDA$
$\displaystyle \Rightarrow \frac { BC }{ AB } =\frac { AD }{ DE } $
$\displaystyle \Rightarrow DE.BC=AD.AB$.
Hence proved.

Multiple choice maths similarity areas of similar figures areas of similar triangles relations between the areas of triangles

If the ratio of the corresponding sides of two similar triangles is 2 : 3, then the ratio of their corresponding altitude is :

  1. 3 : 2

  2. 16 : 81

  3. 4 : 9

  4. 2 : 3
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In two similar triangles, if the corresponding sides are in a particular ratio, then altitudes will also be in the same ratio.
Hence the ratio of the altitudes will be 2 : 3.

Multiple choice maths similarity areas of similar figures areas of similar triangles relations between the areas of triangles

If in $\displaystyle \Delta ABC$ and $\displaystyle \Delta DEF,\frac { AB }{ DE } =\frac { BC }{ FD } $, then they will be similar if :

  1. $\displaystyle \angle B=\angle E$
  2. $\displaystyle \angle A=\angle D$
  3. $\displaystyle \angle B=\angle D$
  4. $\displaystyle \angle A=\angle F$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If two sides of a triangle are proportional to the corresponding two sides in another triangle, and their included angles are equal, then the two triangles are similar by SAS rule.

If $\quad \dfrac { AB }{ DE } = \dfrac { BC }{ FD } $, then for two triangles ABC and DEF to be similar, the included angle must be equal. In this case, the included angles are $\quad \angle B\quad and\quad \angle D$.

Multiple choice maths similarity areas of similar figures areas of similar triangles relations between the areas of triangles

Ratio of areas of two similar triangles is equal to :

  1. ratio of squares of the corresponding altitudes

  2. ratio of squares of corresponding medians.

  3. Either (A) or (B)

  4. (A) and (B) both
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Ratio of areas of two similar triangles is equal to ratio of squares of the corresponding altitudes and ratio of squares of corresponding medians. This means that if the ratio of either altitude or median is given and asked to find ratio of areas , then it will be the ratio of squares of the corresponding altitudes or ratio of squares of corresponding medians.

Multiple choice maths similarity areas of similar figures areas of similar triangles relations between the areas of triangles

If $\Delta ABC\sim \Delta DEF$ such that area of $\Delta ABC$ is $9 cm^2$ and area of $\Delta DEF$ is $16 cm^2$ and $BC=1.8 cm$, then EF is

  1. 2.4 cm

  2. 1.35 cm

  3. 2.1 cm

  4. 3.2 cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$ar(\triangle ABC)=9cm^2,\,ar(\triangle DEF)=16cm^2$ and $BC=1.8cm$

$\triangle ABC\sim\triangle DEF$                [ Given ]

$\Rightarrow$  $\dfrac{ar(\triangle ABC)}{ar(\triangle DEF)}=\dfrac{(BC)^2}{(EF)^2}$                    [ Area of similar triangle theorem ]

$\Rightarrow$  $\dfrac{9}{16}=\dfrac{(1.8)^2}{(EF)^2}$
Taking square root on both sides,

$\Rightarrow$  $\dfrac{3}{4}=\dfrac{1.8}{EF}$

$\Rightarrow$  $EF=\dfrac{1.8\times 4}{3}$

$\Rightarrow$  $EF=\dfrac{7.2}{3}$

$\Rightarrow$  $EF=2.4\,cm$


Multiple choice maths similarity areas of similar figures areas of similar triangles relations between the areas of triangles

The sides of two similar triangles are in the ratio $4:9$ Areas of these triangles are in the ratio

  1. $3 : 5$
  2. $4 : 9$
  3. $81 : 16$
  4. $16 : 81$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If two triangles are similar to each other, then the ratio of the area of this triangle will be equal to the square of the ratio of the corresponding sides of this triangle.

$\therefore$ The ratio between area of these triangle$=\dfrac{(4)^2}{(9)^2}=\dfrac{16}{81}$

Multiple choice maths similarity areas of similar figures areas of similar triangles relations between the areas of triangles

The areas of two similar triangles are $\displaystyle 9\ { cm }^{ 2 }$ and $\displaystyle 16\ { cm }^{ 2 }$, respectively. The ratio of their corresponding heights is

  1. $3 : 4$
  2. $4 : 3$
  3. $2 : 3$
  4. $4 : 5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
In similar traingles: -

${(\dfrac{{h1}}{{h2}})^2} = \dfrac{{S1}}{{S2}}$

Where h1 and h2 are the heights 

and S1, S2 are the areas of similar traingles

${{\rm{(}}\dfrac{{h1}}{{h2}})^2} = \dfrac{{9c{m^2}}}{{16c{m^2}}}$

$\dfrac{{h1}}{{h2}} = \sqrt {\dfrac{9}{{16}}} $

$\dfrac{{h1}}{{h2}} = \dfrac{3}{4}\ or\  {3:4}$