Tag: maths

Questions Related to maths

$\triangle XYZ \sim \triangle DEF$ for the corresponding $XYZ-EFD$ if $mLX:mLY:mLz=2:3:5$ then in $\triangle DEF$_____ is a right angle.

  1. $LD$

  2. $LE$

  3. $LF$

  4. $LE$ or $LF$


Correct Option: A

The ratio of the angles in $\triangle ABC$ is $2 : 3 : 4$. Which one of the following triangles is similar to $\triangle ABC ?$

  1. $ \triangle DEF $ has angles in the ratio $4 : 3 : 2.$

  2. $ \triangle PQR $ has angles in the ratio $1 : 2 : 3.$

  3. $ \triangle LMN $ has angles in the ratio $1 : 1 : 1.$

  4. $ \triangle STW $ has sides in the ratio $1 : 1 : 1.$

  5. $ \triangle XYZ $ has sides in the ratio $4 : 3 : 2.$


Correct Option: A

The length of the sides of $\triangle DEF$ are $4,6,8$  $\triangle DEF \sim \triangle PQR$ for correspondence $DEF \leftrightarrow QPR$ if the perimeter of $\triangle PQR=36$, then the length of the smallest side of $\triangle PQR$ is_____

  1. $2$

  2. $4$

  3. $6$

  4. $8$


Correct Option: A

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

  1. $1$

  2. $2$

  3. $3 $

  4. $4$


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.

  1. BP = 4 cm

  2. BP = 8 cm

  3. BP = 6 cm

  4. BP = 12 cm


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 $DE : BC: ?$

  1. $1:2$

  2. $2:3$

  3. $4:5$

  4. None of these


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 $\angle B: ?$

  1. $69^{\circ}$

  2. $74^{\circ}$

  3. $84^{\circ}$

  4. $94^{\circ}$


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:

  1. unequal

  2. equal

  3. zero

  4. none of these


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.

  1. similar

  2. congruent

  3. both

  4. none of these


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 $\triangle DEF$.

  1. $36 : 50$

  2. $49 : 50$

  3. $36 : 49$

  4. $1:2$


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
$\dfrac {ar(\triangle ABC)}{ar (\triangle DEF)} = \dfrac {AB^{2}}{DE^{2}} = \dfrac {(1.2)^{2}}{(1.4)^{2}} = \dfrac {36}{49}$