Tag: congruence

Questions Related to congruence

If the area of two similar triangles are equal, then they are

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

  1. equilateral

  2. isosceles

  3. congruent

  4. not congruent

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Correct Option: C
Explanation:

They are congruent.

$Consider\triangle ABC\quad and\triangle PQR$
$ \cfrac { ar(\triangle ABC) }{ ar(\triangle PQR) } =\cfrac { { AB }^{ 2 } }{ { PQ }^{ 2 } } =\cfrac { { AC }^{ 2 } }{ { PR }^{ 2 } } =\cfrac { { BC }^{ 2 } }{ { QR }^{ 2 } } $
$\implies\quad AB=PQ,\quad AC=PR,\quad BC=QR$
$ \therefore The\triangle ABC\quad$ and $\triangle PQR$  are congruent.

Two polygons of the same number of sides are similar if all the corresponding interior angles are:

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

  1. Equal

  2. Proportional 

  3. Congruent

  4. Cannot say

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Correct Option: D
Explanation:

Two polygons of the same number of sides are similar, if: 

(a) Their corresponding angles are equal. 
(b) Their corresponding sides are in the same ratio (Proportional).
Hence, nothing can be said about two given polygons when only the angles are congruent, is known.

Triangle is equilateral with side$A$, perimeter $P$, area $K$ and circumradius $R$ (radius of the circumscribed circle). Triangle is equilateral with side $a$, perimeter $p$, area $k$, and circumradius $r$. If $A$ is different from $a$, then

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

  1. $P : p = R : r$ only sometimes

  2. $P : p = R : r$ always

  3. $P : p = K : k$ only sometimes

  4. $R : r = K : k$ always

  5. $R : r = K : k$ only sometimes

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Correct Option: B
Explanation:

Since the triangles are similar, we have $A:a = P:p = R:r = \sqrt {K}: \sqrt {k}$ always, so that (b) is the correct choice.

If in two triangles, corresponding angles are _______ and their corresponding sides are in the ______ratio and hence the two triangles are similar.

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

  1. equal, same

  2. unequal, same

  3. equal, different

  4. unequal, different

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Correct Option: A
Explanation:

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

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

Show answer
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

Show answer
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. $96\ cm^2$

Show answer
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 $ $Area(\triangle DEF)=\dfrac { 54\times 16 }{ 9 } =96{ cm }^{ 2 }$