[3 marks sum]

Question 13 Marks

(i) Plot the points A (3, 5) and B (-2, -4). Use 1 cm = 1 unit on both the axes.

(ii) A’ is the image of A when reflected in the x-axis. Write down the co-ordinates of A’ and plot it on the graph paper.

(iii) B’ is the image of B when reflected in the y-axis, followed by reflection in the origin. Write down the co-ordinates of B’ and plot it on the graph paper.

(iv) Write down the geometrical name of the figure AA’BB’.

(v) Name the invariant points under reflection in the x-axis.

Answer

(i) The points A (3, 5) and B (-2, -4) can be plotted on a graph as shown.
(ii) A’ = Image of A when reflected in the x-axis = (3, -5)
(iii) C = Image of B when reflected in the y-axis = (2, -4) B’ = Image when C is reflected in the origin = (-2, 4)
(iv) Isosceles trapezium
(v) Any point that remains unaltered under a given transformation is called an invariant. Thus, the required two points are (3, 0) and (-2, 0).

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Question 23 Marks

Points A and B have co-ordinates (3, 4) and (0, 2) respectively. Find the image:

(a) A’ of A under reflection in the x-axis.

(b) B’ of B under reflection in the line AA’.

(d) B” of B under reflection in the line AA”.

Answer

(a) A’ = Image of A under reflection in the x-axis = (3, -4)
(b) B’ = Image of B under reflection in the line AA’ = (6, 2)
(c) A” = Image of A under reflection in the y-axis = (-3, 4)
(d) B” = Image of B under reflection in the line AA” = (0, 6)

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Question 33 Marks

A point P (a, b) is reflected in the x-axis to P’ (2, -3). Write down the values of a and b. P” is the image of P, reflected in the y-axis. Write down the co-ordinates of P”. Find the co-ordinates of P”’, when P is reflected in the line, parallel to y-axis, such that x = 4.

Answer

A point P (a, b) is reflected in the x-axis to P’ (2, -3).
We know Mx (x, y) = (x, -y)
Thus, co-ordinates of P are (2, 3). Hence, a = 2 and b = 3.
P” = Image of P reflected in the y-axis = (-2, 3)
P”’ = Reflection of P in the line (x = 4) = (6, 3)

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Question 43 Marks

A point P (-2, 3) is reflected in line x = 2 to point P’. Find the coordinates of P’.

Answer

The line x = 2 is a line parallel to y-axis and at a distance of 2 units from it.
Mark point P (-2, 3).
From P, draw a straight line perpendicular to line CD and produce. On this line mark a point P’ which is at the same distance to the right of CD as P is to the left of it.
The co-ordinates of P’ are (6, 3).

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Question 53 Marks

The points P (4, 1) and Q (-2, 4) are reflected in line y = 3. Find the co-ordinates of P’, the image of P and Q’, the image of Q.

Answer

The line y = 3 is a line parallel to x-axis and at a distance of 3 units from it.
Mark points P (4, 1) and Q (-2, 4).
From P, draw a straight line perpendicular to line CD and produce. On this line mark a point P’ which is at the same distance above CD as P is below it.
The co-ordinates of P’ are (4, 5). Similarly, from Q, draw a line perpendicular to CD and mark point Q’ which is at the same distance below CD as Q is above it.
The co-ordinates of Q’ are (-2, 2).

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Question 63 Marks

Attempt this question on graph paper.

(a) Plot A (3, 2) and B (5, 4) on graph paper. Take 2 cm = 1 unit on both the axes.

(b) Reflect A and B in the x-axis to A’ and B’ respectively. Plot these points also on the same graph paper.

(i) the geometrical name of the figure ABB’A’;

(ii) the measure of angle ABB’;

(iii) the image of A” of A, when A is reflected in the origin.

(iv) the single transformation that maps A’ to A”.

Answer

(c)
(i) From graph, it is clear that ABB’A’ is an isosceles trapezium.
(ii) The measure of angle ABB’ is 45°.
(iii) A” = (-3, -2)
(iv) Single transformation that maps A’ to A” is the reflection in y-axis.

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Question 73 Marks

Use a graph paper for this question: (Take 2cm = 1 unit on both x and y axes) (i) Plot the following points: A(0,4), B(2,3), C(1,1) and D(2,0).

(ii) Reflect points B, C, D on the y-axis and write down their coordinates. Name the images as B', C', D' respectively.

(iii) Join the points A, B, C, D, D', C', B' and A in order, so as to form a closed figure. Write down the equation to the line about which if this closed figure obtained is folded, the two parts of the figure exactly coincide.

Answer

(i)Plotting A(0,4), B(2,3), C(1,1) and D(2,0).

(ii) Reflected points B'(-2,3), C'(-1,1) and D'(-2,0).
(iii) The figure is symmetrical about x = 0

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Question 83 Marks

Use graph paper for this question.

(Take 2 cm = 1 unit along both side x-axis and y-axis.)

Plot the points O(0,0), A(-4, 4), B(-3, 0) and C(0, -3).

Reflect points A and B on the y-axis and name them A' and B' respectively. Write down their co-ordinates.
Name the figure OABCB'A'.
iii. State the line of symmetry of this figure. [Now symmetry is not in course]

Answer

1. $A^{\prime}=(4,4) A N D B^{\prime}=(3,0)$
2. The figure is an arrow head.
3. iii. The $y$-axis i.e. $x=0$ is the line of symmetry of figure $O A B C B^{\prime} A^{\prime}$.

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Question 93 Marks

Using a graph paper, plot the point A (6, 4) and B (0, 4).

(a) Reflect A and B in the origin to get the image A’ and B’.

(b) Write the co-ordinates of A’ and B’.

(d) Find its perimeter.

Answer

(a)

(b) Co-ordinates of A'=(-6,-4)
Co-ordinates of B'=(0,-4)
(c) ABA'B' is parallelogram.
(d) In ABA'B',BB'=8 units, A'B'=6 units
$\therefore B A \prime=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$ units
$\Rightarrow B ı A=10$ units
$A B=A^{\prime} B^{\prime}=6$ units
∴ the perimeter of ABA'B'=AB+BA'+A'B'+B'A
=6+10+6+10
=32 units

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Question 103 Marks

The triangle $A B C$, where $A$ is $(2,6), B$ is $(-3,5)$ and $C$ is $(4,7)$, is reflected in the $y$-axis to triangle $A^{\prime} B^{\prime} C^{\prime}$. Triangle $A^{\prime} B^{\prime} C^{\prime}$ is then reflected in the origin to triangle $A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}$.
i. Write down the co-ordinates of $A ^{\prime \prime}, B ^{\prime \prime}$ and $C ^{\prime \prime}$.
ii. Write down a single transformation that maps triangle $A B C$ onto triangle $A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}$.

Answer

(i) Reflection in $y$-axis is given by $M_y(x, y)=(-x, y) \therefore A^{\prime}=$ Reflection of $A(2,6)$ in $y$-axis $=(-2,6)$
Similarly, $B^{\prime}=(3,5)$ and $C^{\prime}=(-4,7)$
Reflection in origin is given by $M_O(x, y)=(-x,-y)$
$\therefore A ^{\prime \prime}=$ Reflection of $A ^{\prime}(-2,6)$ in origin $=(2,-6)$
Similarly, $B^{\prime \prime}=(-3,-5)$ and $C^{\prime \prime}=(4,-7)$
(ii) A single transformation which maps triangle $A B C$ to triangle $A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}$ is reflection in $x$-axis.

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Question 113 Marks

A (1, 1), B (5, 1), C (4, 2) and D (2, 2) are vertices of a quadrilateral. Name the quadrilateral ABCD. A, B, C, and D are reflected in the origin on to A’, B’, C’ and D’ respectively. Locate A’, B’, C’ and D’ on the graph sheet and write their co-ordinates .

Are D, A, A’ and D’ collinear?

Answer

Quadrilateral ABCD is an isosceles trapezium.
Co-ordinates of A’, B’, C’ and D’ are A'(-1, -1), B'(-5, -1), C'(-4, -2) and D'(-2, -2) respectively.
It is clear from the graph that D, A, A’ and D’ are collinear.

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Question 123 Marks

The point P (3, 4) is reflected to P’ in the x-axis; and O’ is the image of O (the origin) when reflected in the line PP’.

Write:

(i) the co-ordinates of P’ and O’.

(ii) the length of the segments PP’ and OO’.

(iii) the perimeter of the quadrilateral POP’O’.

(iv) the geometrical name of the figure POP’O’.

Answer

(i) Co-ordinates of P’ and O’ are (3, -4) and (6, 0) respectively.
(ii) PP’ = 8 units and OO’ = 6 units.
(iii) From the graph it is clear that all sides of the quadrilateral POP’O’ are equal.
In right Δ PO’Q, PO’ $=\sqrt{(4)^2+(3)^2}=5$ units
So, perimeter of quadrilateral POP’O’ = 4 PO’ = 4 × 5 units = 20 units
(iv) Quadrilateral POP’O’ is a rhombus.

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Question 133 Marks

The point $P (5, 3)$ was reflected in the origin to get the image $P’.(a)$ Write down the co-ordinates of P’.
(b) If M is the foot if the perpendicular from P to the x-axis, find the co-ordinates of M.
(c) If N is the foot if the perpendicular from P’ to the x-axis, find the co-ordinates of N.
(d) Name the figure PMP’N.
(e) Find the area of the figure PMP’N.

Answer

(a) Co-ordinates of $P’ = (-5, -3)$
(b) Co-ordinates of $M = (5, 0)$
(c) Co-ordinates of $N = (-5, 0)$
(d) PMP’N is a parallelogram.
$\text { (e) Are of PMP'N }=\operatorname{ar}(\triangle PMN )+\operatorname{ar}\left(\triangle MNP ^{\prime}\right)$
$=\frac{1}{2} \times 10 \times 3+\frac{1}{2} \times 10 \times 3$
$=15+15$
$=30 \text { sq. units }$

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Question 143 Marks

On a graph paper, plot the triangle ABC, whose vertices are at points A (3, 1), B (5, 0) and C (7, 4).

On the same diagram, draw the image of the triangle ABC under reflection in the origin O (0, 0).

Answer

The graph shows triangle ABC and triangle A’B’C’ which is obtained when ABC is reflected in the origin

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Mathematics [3 marks sum]

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