Tag: introduction to similar triangles

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

Is the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians?

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides, which is also equal to the square of the ratio of their corresponding medians.

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

The areas of two similar triangles $\triangle{ABC}$ and $\triangle{DEF}$ are $144\ cm^{2}$ and $81\ cm^{2}$ respectively. If the longest side of larger $\triangle{ABC}$ be $36\ cm$, then, the largest side of the similar triangle $\triangle{DEF}$ is

  1. $20\ cm$
  2. $26\ cm$
  3. $27\ cm$
  4. $30\ cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, the ratio of their sides is the square root of the ratio of their areas, which is sqrt(144/81) = 12/9 = 4/3. Setting up the proportion 36/x = 4/3 yields x = 27 cm for the smaller triangle's corresponding side.

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

The correspondence $ABC\rightarrow PQR$ is a similarity in $\Delta ABC$ and $\Delta PQR$. If the perimeter of $\Delta ABC$ is $24$ and the perimeter of $\Delta PQR$ is $40$, then $AB=PQ=$

  1. $4:3$
  2. $3:4$
  3. $5:3$
  4. $3:5$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For similar triangles, the ratio of any pair of corresponding sides is equal to the ratio of their perimeters. Given the perimeter of ABC is 24 and PQR is 40, the ratio is 24/40, which simplifies to 3/5. Thus, the ratio AB/PQ equals 3/5.

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, $\angle B = \angle D = 74^{\circ}$