Tag: congruence

Quizzes (1) Questions (67)

Questions Related to congruence

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

It is given that $\Delta ABC \sim \Delta PQR$ with $\dfrac{BC}{QR} = \dfrac{1}{3}$. Then $\dfrac{ar (\Delta PQR)}{ar (\Delta ABC)}$ is equal to

  1. $9$
  2. $3$
  3. $\dfrac{1}{3}$
  4. $\dfrac{1}{9}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
 If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.

Since, $\Delta ABC \sim \Delta PQR$

$\therefore \dfrac{ar (\Delta PQR)}{ar (\Delta ABC)} = \dfrac{PR^2}{AC^2} = \dfrac{QR^2}{BC^2} = \dfrac{9}{1} =9 \ \ \ ..........  \left [ \therefore \dfrac{QR}{BC} = \dfrac{3}{1} \right ]$
Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

$CM$ and $RN$ are respectively the medians of $\triangle {ABC}$ and $\triangle{PQR}$. If $\triangle {ABC}\sim \triangle{PQR}$, then
  $\cfrac{CM}{RN}=\cfrac{AB}{PQ}$

  1. True

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

In similar triangles, the ratio of corresponding medians is equal to the ratio of corresponding sides. Thus, CM/RN = AB/PQ is a true statement.

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

In a square $ABCD$, the bisector of the angle $BAC$ cut $BD$ at $X$ and $BC$ at $Y$ then triangles $ACY, ABX$ are similar.

  1. True

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

In square ABCD, angle BAC = 45 degrees. The bisector of BAC makes angles of 22.5 degrees. Through geometric properties and angle chasing, it can be shown that triangles ACY and ABX are similar.

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

Assume that, $\Delta RST \sim \Delta XYZ$. Complete the following statement.


$\displaystyle \frac{RT}{XY} = \frac{- -}{YZ}, \frac{RS}{XY} = \frac{ST}{- -}, \frac{XY}{ - -} = \frac{YZ}{ST}$

  1. ST, YZ, RT

  2. ST, YZ, RS

  3. YT, YS, RZ

  4. ST, YZ, RZ
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given triangle RST similar to triangle XYZ, the ratios of corresponding sides are equal: RS/XY = ST/YZ = RT/XZ. The provided option B correctly completes the ratios.

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles
It is given that $\triangle FED\sim \triangle STU$. Is it true to say that $\cfrac{DE}{UT}=\cfrac{EF}{TS}$? 

  1. Yes

  2. No

  3. Cannot say

  4. None
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\triangle FED \sim \triangle STU$
The corresponding sides of both the triangles are $F\leftrightarrow S$, $E\leftrightarrow T$, $D\leftrightarrow  U$. With this correspondence,
$\cfrac{EF}{ST}=\cfrac{DE}{TU}$

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

Consider the following statements:
(1) If three sides of triangle are equal to three sides of another triangle, then the triangles are congruent.
(2) If three angles of a triangle are respectively equal to three angles of another triangle, then the two triangles are congruent.

Of these statements,

  1. $(1)$ is correct and $(2)$ is false
  2. Both $(1)$ and $(2)$ are false
  3. Both $(1)$ and $(2)$ are correct
  4. $(1)$ is false and $(2)$ is correct
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If three sides of triangle are equal to other three sides of the triangle, then the two triangles are congruent by SSS rule.

If three angles of the triangle are equal to other three sides of the triangle, then the two triangles are similar by AAA rule but not congruent.

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

If in trianges $ABC$ and $DEF$, $\cfrac{AB}{DE}=\cfrac{BC}{FD}$, then they will be similar, when:

  1. $\angle B=\angle E$
  2. $\angle A=\angle D$
  3. $\angle B=\angle D$
  4. $\angle A=\angle F$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In $\triangle ABC$ and $\triangle DEF$,
$\dfrac{AB}{DE} = \dfrac{BC}{FD}$ (Given)

The angle between these sides are $\angle B$ and $\angle D$. Thus, If the containing angles are equal. The triangles will be similar..

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

In $\triangle PQR,$ $PQ=4$ cm, $QR=3$ cm, and $RP=3.5$ cm. $\triangle DEF$ is similar to $\triangle PQR.$ If $EF=9$ cm, then what is the perimeter of $\triangle DEF: ?$

  1. $10.5$ cm
  2. $21$ cm
  3. $31.5$ cm

  4. Cannot be determined as data is insufficient
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$PQ = 4, QR = 3$ and $RP = 3.5$
Also, $EF = 9$
Now, perimeter of $\triangle PQR = PQ + QR + RP = 4 + 3 + 3.5 = 10.5$
Given, $\triangle DEF \sim \triangle PQR$
$\dfrac{EF}{QR} = \dfrac{Perimeter(\triangle DEF)}{Perimeter(\triangle PQR)}$
$\dfrac{9}{3} = \dfrac{Perimeter(\triangle DEF)}{10.5}$
Perimeter $(\triangle DEF) = 31.5$ cm

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

The perimeter of two similar triangles are $24$ cm and $16$ cm, respectively. If one side of the first triangle is $10$ cm, then the corresponding side of the second triangle is

  1. $9$ cm
  2. $\dfrac{20}3$ cm
  3. $\dfrac{16}3$ cm
  4. $5$ cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
In similar triangles, ratio of the sides is equal to the ratio of the perimeters.
Thus, $\dfrac{p _1}{p _2} = \dfrac{s _1}{s _2}$
$\dfrac{24}{16} = \dfrac{10}{s _2}$
$s _2 = \dfrac{20}{3}$
Thus, side of the other triangle is $\dfrac{20}{3}$ cm.