Tag: congruence

Questions Related to congruence

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

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

  1. $1$

  2. $2$

  3. $3 $

  4. $4$

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Correct Option: B
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.

In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point in side AC such that AP = 6 cm. Prove that $\Delta\,BPC$ is similar to $\Delta\,ABC$. Also, find the length of BP.

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

  1. BP = 4 cm

  2. BP = 8 cm

  3. BP = 6 cm

  4. BP = 12 cm

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Correct Option: A
Explanation:

Given: $\triangle ABC$, $AB = AC = 8$, $BC = 4$ and $AP = 6$

In $\Delta\,ABC$,
$\displaystyle\,\frac{AB}{BC}\,=\,\frac{8}{4}\,=\,2$,
In $\Delta\,BPC$,
$\displaystyle\,\frac{BC}{CP}\,=\,\frac{4}{2}\,=\,2$

Now, in $\triangle ABC$ and $\triangle BPC$
$\displaystyle\,\dfrac{AB}{BC}\,= \displaystyle\,\dfrac{BC}{CP}$
$\angle\,ABC\,=\,\angle\,C.$
Therefore, by SAS, $\Delta\,ABC \sim \Delta\,BPC$

Thus, $\dfrac{AB}{BP} = \dfrac{AC}{BC}$


$\dfrac{8}{BP} = \dfrac{8}{4}$
$BP = 4$ cm

In the given figure, $DE$ is parallel to $BC$ and the ratio of the areas of $\triangle ADE$ and trapezium $BDEC$ is $4:5.$ What is $DE : BC: ?$

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

  1. $1:2$

  2. $2:3$

  3. $4:5$

  4. None of these

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Correct Option: B

If in $\triangle $s $ABC$ and $DEF,$ $\angle A=\angle E=37^{\circ}, AB:ED=AC:EF$ and $\angle F=69^{\circ},$ then what is the value of $\angle B: ?$

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

  1. $69^{\circ}$

  2. $74^{\circ}$

  3. $84^{\circ}$

  4. $94^{\circ}$

Show answer
Correct Option: B
Explanation:

In $\triangle ABC$ and $\triangle DEF$
$\angle A = \angle E =  37^{o}$
$\dfrac{AB}{ED} = \dfrac{AC}{EF}$
Thus, $\triangle ABC \sim \triangle EDF$ ....... (By SAS rule)
Thus, $\angle B = \angle D$

Now, $\triangle DEF$
$\angle D + \angle E + \angle F = 180$
$\angle D + 37 + 69 = 180$
$\angle D = 74^{\circ}$
Hence, $\angle B = \angle D = 74^{\circ}$

If two triangles are similar then, ratio of corresponding sides are:

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

  1. unequal

  2. equal

  3. zero

  4. none of these

Show answer
Correct Option: B
Explanation:
Similar triangles have $:$
$i)$ All their angles equal
$ii)$ Corresponding sides have the same ratio

So, option $B$ is correct. 

Two equilateral triangles with side $4 \ cm$ and $6 \ cm$ are _____ triangles.

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

  1. similar

  2. congruent

  3. both

  4. none of these

Show answer
Correct Option: A
Explanation:

Any two equilateral triangles are similar by SSS criteria..
$SSS$ similarity states that if the lengths of the corresponding sides of two triangles are proportional, then the triangles must be similar.
Two equilateral triangles with side $4 \ cm$ and $6 \ cm$ are similar triangles by $SSS$ similarlty. 

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 $\triangle DEF$.

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

  1. $36 : 50$

  2. $49 : 50$

  3. $36 : 49$

  4. $1:2$

Show answer
Correct Option: C
Explanation:

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

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.

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

  1. $8\ cm$

  2. $4\ cm$

  3. $3\ cm$

  4. $16\ cm$

Show answer
Correct Option: A
Explanation:

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

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

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

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

$\Rightarrow DE = 8\ cm$

Which of the following is/are the property of similar figures?

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

  1. Corresponding angles are congruent.

  2. Corresponding sides are in the same ratio.

  3. Both A and B

  4. None

Show answer
Correct Option: C
Explanation:

Shape can be different for similar figures be it circle, be it rectangles but if corresponding angles are equal and sides or radius in case of circle are in equal  ratio, then the corresponding two figures are similar.

$\displaystyle \Delta ABC$ and $\displaystyle \Delta DEF$ are two similar triangles such that $\displaystyle \angle A={ 45 }^{ \circ  },\angle E={ 56 }^{ \circ  }$, then $\displaystyle \angle C$ =___.

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

  1. $\displaystyle { 56 }^{ \circ  }$

  2. $\displaystyle { 45 }^{ \circ  }$

  3. $\displaystyle { 101 }^{ \circ  }$

  4. $\displaystyle { 79 }^{ \circ  }$

Show answer
Correct Option: D
Explanation:

$\Delta ABC \sim \Delta DEF$        ...Given

$\Rightarrow \angle A = \angle D$                 ...C.A.S.T.
$\Rightarrow \angle B = \angle E$                 ...C.A.S.T.
$\Rightarrow \angle C = \angle F$                 ...C.A.S.T.
$\therefore \angle B = \angle E = 56^o$
In $\Delta ABC$,
$\angle A + \angle B + \angle C = 180^o$        ....Angle sum property of triangles
$\Rightarrow 45^o+56^o+\angle C = 180^o$
$\Rightarrow \angle C = 79^o$