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Mathematics [3 marks sum]

Reflection · ICSE STD 10 (ICSE - English Medium). 19 questions.

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[3 marks sum]

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19 questions · timed · auto-graded

Question 13 Marks
Point A (5, 1) on reflection on X- axis is mapped as A’. Also A on reflection on Y- axis is mapped as A”.
(i) Write the co-ordinates of A’.
(ii) Write the co-ordinates of A”.
(iii) Calculate the distance A’ A”.
(iv) On which coordinate axis does the middle point M of A” A’ lie?
Answer
(i) A' → (5, -1).
(ii) A'' → (-5, -1).

(iii) Distance A'A'' = 5 + 5 = 10.
(iv) Miles on Y-axis.
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Question 23 Marks
A point P(4, – 1) is reflected to P’ in the line y = 2 followed by the reflection to P” in the line x = -1. Find :
(i) The co-ordinates of P’.
(ii) The co-ordinates of P”.
(iii) The length of PP’.
(iv) The length of P’P”.
Answer

(i) P' → (4, 5).
(ii) P'' → (-6, -5).
(iii) PP' = 6 units.
(iv) P'P'' = 10 units.
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Question 33 Marks
P, Q have co-ordinates (-1, 2) and (6, 3) respectively. Reflect P on the X-axis to P’. Find:
(i) The co-ordinate of P’
(ii) Length of P’Q.
(iii) Length of PQ.
(iv) Is P’Q = PQ?
Answer

(i) $P^{\prime} \rightarrow(-1-2)$.
(ii) $P^{\prime} Q=\sqrt{(6+1)^2+(3+2)^2}$
$=\sqrt{49+25}$
$=\sqrt{74} .$
(iii) $P Q=\sqrt{(6+1)^2+(3-2)^2}$
$=\sqrt{49+1}$
$=\sqrt{50} $
(iv) No. $\left(P^{\prime} Q \neq P Q\right)$
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Question 43 Marks
Using a graph paper, plot the points $A(6,4)$ and $B(0,4)$.
(i) Reflect $A$ and $B$ in the origin to get the images $A^{\prime}$ and $B^{\prime}$.
(ii) Write the co-ordinates of $A ^{\prime}$ and $B ^{\prime}$.
(iii) State the geometrical name for. the figure $A B A^{\prime} B^{\prime}$.
(iv) Find its perimeter.
Answer
(i)

(ii) $A' (-6, -4)$ and $B' (0, -4)$
(iii) $ABA'B$' is a parallelogram.
(iv) From the figure $AB = 6, BB' = 8, A'B' = 6$
In $\triangle ABB', (AB)2$
$= (AB)^2 + (BB')^2$
$= 6^2 + 8^2$
$= 100$
$\therefore AB' = 10 = A'B' ...\{ABA'B'$ is a parallelogram$\}$
$\therefore$ Perimeter of $ABA'B'$
$= AB + BA' + A'B' + B'A$
$= 6 + 10 + 6 + 10$
$= 32$ units.
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Question 53 Marks
Use graph paper to answer the following questions. (Take 2 cm = 1 unit on both axis).
(i) Plot the points A (- 4, 2) and B (2, 4).
(ii) A’ is the image of A when reflected in the y-axis. Plot it on the graph paper and write the coordinates of A’.
(iii) B’ is the image of B when reflected in the line AA’. Write the coordinates of B’.
(iv) Write the geometric name of the figure ABA’B’.
(v) Name a line of symmetry of the figure formed.
Answer
(i) On the graph.
(ii) Co-ordinates of A' = (4, 2).
(iii) Co-ordinates of B' = (2, 0).
(iv) Geometric name of Fig. ABA'B' is kite.
(v) Line of symmetry = AA'.
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Question 63 Marks
Use graph paper to answer this question:
(i) Plot the points A (4,6) and B (1, 2).
(ii) A’ is the image of A when reflected in X-axis,
(iii) B’ is the image of B when B is reflected in the line AA’.
(iv) Give the geometrical name for the figure ABA’B’.
Answer

(i) In the graph paper.
(ii) A (4, 6) → A' (4, -6)
(iii) B (1, 2) → B' (7, 2)
(iv) ABA'B' is a kite.
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Question 73 Marks
Use graph paper for this question.
The point P (5, 3) was reflected in the origin to get the image P’.
(i) Write down the co-ordinates of P’.
(ii) If M is the foot of the perpendicular from of P to the X-axis, find the co-ordinates of M.
(iii) If N is the foot of the perpendicular from of P’ to the X-axis, find the co-ordinates of N.
(iv) Name the figure PMP’N.
(v) Find the area of die figure PMP’N.
Answer

(i) P' (-5, -3).
(ii) M (5, 0).
(iii) N (-5, 0).
(iv) The figure PMP'N is a parallelogram.
(v) The area of figure PMP'N
$=\frac{1}{2} \times 10 \times 3+\frac{1}{2} \times 10 \times 3$
= 15 + 15
= 30 sq. units.
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Question 83 Marks
Using graph paper and taking 1 cm = 1 unit along both x-axis and y-axis.
(i) Plot the points A (- 4, 4) and B (2, 2).
(ii) Reflect A and B in the origin to get the images A’ and B’ respectively.
(iii) Write down the co-ordinates of A’ and B’.
(iv) Give the geometrical name for the figure ABA’B’.
(v) Draw and name its lines of symmetry.
Answer
(i),(ii) (In the graph paper)
(iii) A' (4, -4) B' (-2, -2)
(iv) Rhombus
(v) AA', and BB'
Image
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Question 93 Marks
Use a graph paper for this question (take 10 small divisions = 1 unit on both axis).
Plot the points P (3, 2) and Q (-3, -2), from P and Q draw perpendicular PM and QN on the X- axis.
(i) Name the image of P on reflection at the origin.
(ii) Assign, the. special name to the geometrical figure. PMQN and find its area.
(iii) Write the co-ordinates of the point to which M is mapped on reflection in (i) X- axis,
(ii) Y-axis, (iii) origin.
Answer

In the graph paper
(i) Q (-3, -2)
(ii) Parallelgoram;
Area of ΔPMN
$=\frac{1}{2} PM \times MN$
$=\frac{1}{2} \times 2 \times 63$
∴ Area of PMQN
= 2 x ΔPMN
= 2 x 6
= 12 square unit
(iii) Co-ordinates of M(3, 0)
(i) (3, 0), (ii) (-3, 0), (iii) (-3, 0).
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Question 103 Marks
Use graph paper for this question.
The points A (2, 3), B (4, 5) and C (7, 2) are the ,vertices of A ABC.
(i) Write down the coordinates of A’, B’, C’ if Δ A’B’C’ is the image of Δ ABC, when reflected in the origin.
(ii) Write down the co-ordinates of A”, B”, C” if A” B” C” is the image of Δ ABC, when reflected in the x-axis.
(iii) Mention the special name of the quadrilateral BCC” B” and find its area.
Answer
(i) Reflection in origin
$( x , y ) \xrightarrow{ M _0}=(- x ,- y )$
$A (2,3) \xrightarrow{ M _0}= A ^{\prime}(-2,-3)$
$B (4,5) \xrightarrow{ M _0}= B ^{\prime}(-4,-5)$
$C (7,2) \xrightarrow{ M _0}= C ^{\prime}(-7,-2)$

(ii) Now A, B, C is reflected in X axis.
Reflection in X axis
$( x , y ) \xrightarrow{ Mx }=( x ,- y )$
$A (2,3) \xrightarrow{ Mx }= A / \prime(2,-3)$
$3(4,5) \xrightarrow{ Mx }= B / \prime(4,-5)$
$C (7,2) \xrightarrow{ Mx }= C \prime(7,-2)$
(iii) BCC"B" is an isosceles trapezium.
CD = 7 - 4 = 3
CC" = 2 + 2 = 4 and
BB" = 5 + 5 = 10
Area of Trapexium $=\frac{1}{2}\left(C^{\prime \prime}+B^{\prime \prime}\right) \times C D$
$=\frac{1}{2}(4+10) \times 3$
$=\frac{1}{2} \times 14 \times 3$
$=21 \text { sq. unit }$
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Question 113 Marks
The point P(3, 4) is reflected to P’ in the x-axis and O’ is the image of O (the Origin) in the line PP’ Find :
(i) The coordinates of P’ and O’.
(ii) The length of segment PP’ and OO’.
(iii) The perimeter of the quadrilateral POP’O’
(iv) What is the special name of the quadrilateral POP’O’.
Answer
(i) P' (3, -4), O'(6, 0)

(ii) PP'
$=\sqrt{(3-3)^2+(-4-4)^2}$
$=\sqrt{64}$
= 8 units
OO'
$=\sqrt{(0-6)^2+(0-0)^2}$
$=\sqrt{36}$
= 6 units.
(iii) Perimeter
= 5 + 5 + 5 + 5
= 20 units.
(iv) Rhombus.
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Question 123 Marks
The image of triangle $O X Y$ under reflection in the origin, $O$ is the triangle $O X_1 Y_1$, where $X_1(-3,-4)$ is the image of $X$ and $Y_1$, $(0,-5)$ is the image of $Y$.
(i) Draw a diagram to represent this information and write down the co-ordinates of $X$ and $Y$.
(ii) What kind of figure is the quadrilateral $X Y X_1 Y_1$ ? Give reason for your answer. State, with a reason, whether the figure
$X Y X_1 Y_1$ has any lines of symmetry.
(iii) Find the co-ordinates of $X _2$, the image of X under reflection in the origin followed by reflection on the Y -axis.
(iv) Find the co-ordinates of $Y _2$, the image of Y under reflection on the X -axis followed by reflection in the origin.
Answer
(i)
$X_1 \rightarrow (-3, -4)$
$X \rightarrow (3, 4)$
$Y_1 \rightarrow (0, -5)$
$Y \rightarrow (0, 5).$​​​​​​​

(ii) Rectangle.
Reasons : $X_1Y_1 = YX$
and $X_1Y_1 = Y_1X$
Also each angle of the quadrilateral $XYX_1Y_1$ is $90^\circ .$
It has two lines of symmetry. These are the perpendicular bisectors of each pair of opposite sides.
$(iii) X_2= (3, -4).$
$(iv) Y_2 = (0, 5).$
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Question 133 Marks
Use a graph paper for this question. (Take $10$ small divisions $=1$ unit on both axis). $P$ and $Q$ have co-ordinates $(0,5)$ $(-2,4)$.
(i) P is invariant when reflected in an axis. Name the axis.
(ii) Find the image of $Q$ on reflection in the axis found in (i).
(iii) $(0, k)$ on reflection in the origin is invariant. Write the value of $k$.
(iv) Write the co-ordinates of the image of $Q$ , obtained by reflecting it in the origin following by reflection in $x$ -axis.
Answer
(i) The axis is y-axis or $x = 0.$
(ii) Image of $'Q'$
$Q' = M_{x = 0}(-2, 4)$
$= (2, 4)$
$(iii) \because M_0 (a, b) = (-a, -b)$
$\therefore M_0 (0, k) = (o, -k)$
$\therefore k = -k$
$\therefore 2k = 0$
$\therefore k = 0$
$(iv) Q'' = MxMoQ$
$= M_xM_0(-2, 4)$
$= Mx (2, -4)$
$= (-2, -4)$​​​​​​​
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Question 143 Marks
Use a graph paper to answer the following questions. (Take 1 cm = 1 unit on both axis):
(i) Plot A (4, 4), B (4, – 6) and C (8, 0), the vertices of a triangle ABC.
(ii) Reflect ABC on the y-axis and name it as A’B’C’.
(iii) Write the coordinates of the images A’, B’ and C’.
(iv) Give a geometrical name for the figure AA’ C’B’ BC.
(v) Identify the line of symmetry of AA’ C’ B’ BC.
Answer
(i) and (ii) see the given graph.
(iii) A' (-4, 4), B' (-4, -6), C' (-8, 0).
(iv) AA' C'' B' BC is a Hexagon.
(v) y-axis is the line of symmetry.
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Question 153 Marks
A point P(a, b) is reflected in the X-axis to P'(2, – 3). Write down the value of a & b. P” is the image of P, when reflected on the Y-axis. Write down the co-ordinates of P” when P is reflected in the line parallel to the Y-axis, such that x = 4.
Answer
Reflection of P (a, b) on the X-axis P' (a, -b)
(i) ∴ P' (a, -b) = P' (2, -3)
+ a = + 2, ∴ a = 2
-b = -3, ∴ b = 3
P (2, 3)

(ii) P'' is the image of P(2, 3) under reflection Y-axis = P''(-2, 3).
(iii) P''' is the image of P(2, 3) under reflection x = 4
⇒ P'''(6, 3).
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Question 163 Marks
Points $(3,0)$ and $(-1,0)$ are invarient points under reflection in the line $L_1 ;$ point $(0,-3)$ and $(0,1)$ are invarient points on reflection in line $L _2$.
(i) Write the equation of the line $L_1$ and $L_2$.
(ii) Write down the images of points $P (3,4)$ and $Q (-5,-2)$ on reflection in $L _1$. Name the images as $P ^{\prime}$ and $Q ^{\prime}$ respectively.
(iii) Write down the images of P and Q on reflection in $L _2$. Name the image as $P ^{\prime \prime}$ and $Q ^{\prime \prime}$ respectively.
Answer
(i) $(3,0)$ and $(-1,0)$ lies on $X$ -axis, so these are invariant under reflection on the $X$ -axis. Hence, $L _1$ lies on X -axis So, equation of line $L_1$, is $y=0$.
$(0,-3)$ and $(0,1)$ lies on $Y$-axis, so these are invariant under reflection on the $Y$-axis. So, equation of line $L_2$ is $y=0$.
$(ii) P' \Rightarrow (3, -4)$
$Q' \Rightarrow (-5, 2).$
$(iii) P'' \Rightarrow (-3, 4)$
$Q'' \Rightarrow (5, -2).$
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Question 173 Marks
(i) Point P(a, b) reflected on the X-axis to P'(5, 2). Write down the value of a and b.
(ii) P” is the image of P when reflected on the Y-axis. Write down the co-ordinates of P”.
(iii) Name a single transformation that maps P’ to P”.
Answer
(i) The value of a = 5 and b = -2.
(ii) Co-ordinates of P'' = (-5, -2).
(iii) (x, y) → (-x, -y).
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Question 183 Marks
Point A (2, -4) is reflected in origin as A’. Point B (- 3, 2) is reflected on X-axis as B’.
(i) Write the co-ordinates of A’.
(ii) Write the co-ordinates of B’.
(iii) Calculate the distance A’B’.
Give your answer correct to 1 decimal place, (do not consult tables).
Answer

(i) A' → (-2, 4).
(ii) B' → (-3, -2).
(iii) Distance A'B'
$=\sqrt{(-3+2)^2+(-2-4)^2}$
$=\sqrt{1+36}$
$=\sqrt{37}$
$=6.1$
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Question 193 Marks
Point A(4, – 1) is reflected as A’ on Y-axis. Point B on refletion on X-axis is mapped as B’ (- 2, 5).
(i) Write the co-ordinates of A’.
(ii) Write the co-ordinates of B.
(iii) Write the co-ordinates of the middle point
M of the segment A’B.
(iv) Write the co-ordinates of the point of reflection A” of A on X-axis.
Answer
Image
(i) $A ^{\prime} \rightarrow(-4,-1)$.
(ii) $B \rightarrow(-2,-5)$.
(iii) $M \rightarrow(-3,-3)$.
(iv) $A^{\prime \prime} \rightarrow(4,1)$.
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[3 marks sum] - Mathematics STD 10 Questions - Vidyadip