Tag: maths

Questions Related to maths

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.

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

  1. $42m$

  2. $48m$

  3. $62m$

  4. $52m$

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Correct Option: A
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

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

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

  1. $24$

  2. $27$

  3. $18$

  4. $8$

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Correct Option: B
Explanation:

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

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

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

  1. True

  2. False

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Correct Option: A
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.

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

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

  1. 3 : 2

  2. 16 : 81

  3. 4 : 9

  4. 2 : 3

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Correct Option: D
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.

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

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

  1. $\displaystyle \angle B=\angle E$

  2. $\displaystyle \angle A=\angle D$

  3. $\displaystyle \angle B=\angle D$

  4. $\displaystyle \angle A=\angle F$

Show answer
Correct Option: C
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 $\quad \angle B\quad and\quad \angle D$.

Ratio of areas of two similar triangles is equal to :

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

  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

Show answer
Correct Option: D
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.

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

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

  1. 2.4 cm

  2. 1.35 cm

  3. 2.1 cm

  4. 3.2 cm

Show answer
Correct Option: A
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,

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

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

$\Rightarrow$  $EF=\dfrac{7.2}{3}$

$\Rightarrow$  $EF=2.4\,cm$


Two similar triangles have

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

  1. equal sides

  2. equal areas

  3. equal angles

  4. None of these

Show answer
Correct Option: C
Explanation:

Two similar triangles have equal angles.

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

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

  1. $3 : 5$

  2. $4 : 9$

  3. $81 : 16$

  4. $16 : 81$

Show answer
Correct Option: D
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$=\dfrac{(4)^2}{(9)^2}=\dfrac{16}{81}$

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

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

  1. $3 : 4$

  2. $4 : 3$

  3. $2 : 3$

  4. $4 : 5$

Show answer
Correct Option: A
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}}} $

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