Tag: similarity of triangles

Questions Related to similarity of triangles

State True or False
If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar.

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

  1. True

  2. False

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Correct Option: A
Explanation:

.The statement is true if in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar by $AA$ similarity criteria.

Which among the following is/are not correct ?

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

  1. The ratios of the areas of two similar triangles is equal to the ratio of their corresponding sides.

  2. The areas of two similar triangles are in the ratio of the corresponding altitudes.

  3. The ratio of area of two similar triangles are in the ratio of the corresponding medians.

  4. If the areas of two similar triangles are equal, then the triangles are congruent.

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Correct Option: A,B,C
Explanation:

The theorem is that the ratio of the areas of two similar triangles is equal to the square of the ratio of the corresponding sides.
In options A, B, and C this condition does not hold.
So option A, B, and C are not true.
But option D is true because if the areas of the similar triangles are equal then the sides will also be equal.
So, the triangles will be congruent by SSS test .

State True or False.
If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar.

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

  1. True

  2. False

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Correct Option: A
Explanation:

If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar by $SAS$ similarity criteria.

Therefore the statement is $True$.

The ratio of the areas of two  similar triangles is equal to the

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

  1. ratio of corresponding medians

  2. ratio of corresponding sides

  3. ratio of the squares of corresponding sides

  4. none of these

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Correct Option: C
Explanation:

The area of triangle is proportional to the square of the side of the triangle.
ratio of areas of two similar triangles= ratio of the squares of corresponding sides 

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 $\triangle PQR$.

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

  1. $16:81$

  2. $9:16$

  3. $81:16$

  4. $16:9$

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Correct Option: A
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 $ $\dfrac { Area(\triangle ABC) }{ Area(\triangle PQR) }$ = $\dfrac{16}{81}$

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 $\triangle DEF$.

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

  1. $40\ cm^2$

  2. $59\ cm^2$

  3. $69\ cm^2$

  4. $96\ cm^2$

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Correct Option: D
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 $ $Area(\triangle DEF)=\dfrac { 54\times 16 }{ 9 } =96{ cm }^{ 2 }$

Two $\triangle sABC $ and DEF are similar. If $ar(DEF)= 243\ cm^2, ar(ABC)=108\ cm^2$ and $BC= 6\ cm$. Find $EF$.

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

  1. $9$

  2. $81$

  3. $3$

  4. $72$

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Correct Option: A
Explanation:
Given:-
$\triangle{ABC} \simeq \triangle{DEF}$
$ar \left( DEF \right) = 243 {cm}^{2}$
$ar \left( ABC \right) = 108 {cm}^{2}$
$BC = 6 cm$

To Find:- $EF = ?$

As we know that,
$\because \; \triangle{ABC} \simeq \triangle{DEF}$

$\cfrac{ar \left( \triangle{ABC} \right)}{ar \left( \triangle{DEF} \right)} = {\left( \cfrac{AB}{DE} \right)}^{2} = {\left( \cfrac{BC}{EF} \right)}^{2} = {\left( \cfrac{AC}{DF} \right)}^{2}$

$\therefore \; \cfrac{ar \left( \triangle{ABC} \right)}{ar \left( \triangle{DEF} \right)} = {\left( \cfrac{BC}{EF} \right)}^{2}$

$\Rightarrow \; \cfrac{108}{243} = \cfrac{{6}^{2}}{{EF}^{2}}$

$\Rightarrow \; {EF}^{2} = \cfrac{243}{108} \times 36$

$\Rightarrow \; EF = \sqrt{81}$

$\Rightarrow \; EF = 9$

Hence, the correct answer is $9$.

$\Delta ABC$ and $\Delta DEF$ are similar and $\angle A=40^\mathring \ ,\angle E+\angle F=$

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

  1. $140$

  2. $40$

  3. $80$

  4. $180$

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Correct Option: A
Explanation:

Since the triangles are similar.

$\angle A=\angle D$
$\angle D=40^{\circ}$
In triangle $\Delta DEF$
$\angle D+\angle E+\angle F=180^{\circ}$
$\angle E+\angle F=180^{\circ}-40^{\circ}=140^{\circ}$

STATEMENT - 1 : If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar.
STATEMENT - 2 : If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.

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

  1. Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 1

  2. Statement - 1 is True, Statement - 2 is True : Statement 2 is NOT a correct explanation for Statement - 1

  3. Statement - 1 is True, Statement - 2 is False

  4. Statement - 1 is False, Statement - 2 is True

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Correct Option: B
Explanation:

 If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar.

If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar.
Both the statements are correct but $2$ is not the reason for $1$
If two corresponding angles are equal then the third corresponding become also equal , so the triangles are similar.
Option $B$ is correct

If $\triangle ABC $ and $BDE$ are similar triangles such that $2AB = DE$ and $BC= 8$ cm, then $EF$ is

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

  1. $16$ cm

  2. $17$ cm

  3. $4$ cm

  4. $8$ cm

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Correct Option: A