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Questions Related to maths

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

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 $4 : 3 : 2.$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Similar triangles must have the same ratio of angles. The ratio 2:3:4 is equivalent to 4:6:8 or any scalar multiple, but the order of the ratio matters for similarity. Option A provides the same ratio 4:3:2, which represents the same set of interior angles as 2:3:4.

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

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. $8$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The sides of triangle DEF are 4, 6, and 8, giving a perimeter of 4 + 6 + 8 = 18. Triangle PQR is similar with a perimeter of 36, meaning the scale factor from DEF to PQR is 36/18 = 2. Multiplying the smallest side of DEF (which is 4) by this scale factor gives 4 * 2 = 8.

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

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. $4$
Reveal answer Fill a bubble to check yourself
B Correct answer
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.
Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

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
Reveal answer Fill a bubble to check yourself
A Correct answer
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

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

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
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The ratio of area(ADE) to area(ABC) is 4/(4+5) = 4/9. Since the ratio of areas of similar triangles is the square of the ratio of their corresponding sides, DE/BC = sqrt(4/9) = 2/3.

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

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. $94^{\circ}$
Reveal answer Fill a bubble to check yourself
B Correct answer
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, $\angle B = \angle D = 74^{\circ}$

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

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

  1. similar

  2. congruent

  3. both

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
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. 

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

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. $1:2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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