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#### Page No 128:

#### Question 1:

What is the speciality of the *y*-coordinates of points on the *x*-axis?

#### Answer:

For every point lying on the *x*-axis, *y*-coordinates are always zero.

#### Page No 128:

#### Question 2:

What is the speciality of the *x*-coordinates of points on the *y*-axis?

#### Answer:

For every point lying on the *y*-axis, *x-*coordinates are always zero.

#### Page No 128:

#### Question 3:

What are the coordinates of the origin?

#### Answer:

The coordinates of the origin are (0, 0).

#### Page No 129:

#### Question 2:

In the figure below, ABCD is a rectangle with the origin O as its centre and sides parallel to the axis.

What are the coordinates of B, C, D?

#### Answer:

Given: *ABCD* is a rectangle with sides parallel to the axes.

∴ *AB *= *DC *and *BC* = *AD *

It can be observed that the coordinate axes have divided the given rectangle into four congruent rectangles.

The coordinates of point *A* are (3, 2).

Therefore, the perpendicular distance of point *A* from *x*-axis and *y*-axis is 2 units and 3 units respectively.

Thus, the perpendicular distance of point *B,* point *C, *and* *point *D* from *x*-axis and *y*-axis would be 2 units and 3 units respectively.

Point *B *lies in the second quadrant.

Therefore, the *x*-coordinate of point *B* is −3 and the *y*-coordinate is 2.

So, the coordinates of point *B* are (−3, 2).

Point *C* lies in the third quadrant.

Therefore, the *x*-coordinate of point *C* is −3 and the *y*-coordinate is −2.

So, the coordinates of point *C* are (−3, −2).

Point *D* lies in the fourth quadrant.

Therefore, the *x*-coordinate of point *D* is 3 and the *y*-coordinate is −2.

So, the coordinates of point *D* are (3, −2).

#### Page No 129:

#### Question 1:

Find the coordinates of the other three vertices of the rectangle in the figure below:

The unit of length used in this centimetres. What is actual width and height of this rectangle?

#### Answer:

Given: *ABCO* is a rectangle.

We know that in a rectangle, opposite sides are equal.

∴ *AB* = *OC* and* AO *= *BC*

The coordinates of point *B* are (4, 3).

Therefore, the perpendicular distance of point* B *from *x*-axis is 3 units and from *y*-axis is 4 units.

Point *A* of the given rectangle is lies on the *y*-axis. Therefore, its *x*-coordinate is 0.

Also, the perpendicular distance of point *A* from the origin = Perpendicular distance of point* B *from the *x*-axis = 3 units.

Therefore, the *x*-coordinate of point *A* is 0 and the *y*-coordinate is 3.

So, the coordinates of point *A* are (0, 3).

Point* O* is the origin of the rectangle. So, the coordinates of point *O* are (0, 0).

Point *C *of the given rectangle is lies on the *x*-axis. Therefore, its *y*-coordinate is 0.

The perpendicular distance of point *C *from the origin* *= Perpendicular distance of point* B *from the *y*-axis = 4 units.

Therefore, the *x*-coordinate of point *C* is 4 and the *y*-coordinate is 0.

So, the coordinates of point *C* are (4, 0).

Hence, the coordinates of the other three vertices of the given rectangle are (0, 3), (0, 0) and (4, 0).

Unit length used in the figure =

∴ Actual width of the rectangle =

Actual height of the rectangle =

#### Page No 130:

#### Question 1:

In the figure below, ABCD is a square. Find the coordinates of B, C, D.

#### Answer:

It is given that *ABCD* is a square.

We know that diagonals of a square are equal and are perpendicular bisectors of each other.

∴ *AO* = *BO *=* DO *= *CO*

The coordinates of point *A* are (2, 0).

Therefore, the perpendicular distance of point A from *y*-axis is 2 units.

∴ *AO* = 2 units

⇒ *BO *=* DO *= *CO* = 2 units

As the length of *BO* is 2 units, the *y*-coordinate of point *B* is 2.

Also, as point *B* is lies on the *y*-axis, the *x*-coordinate of point *B* is 0.

So, the coordinates of point *B *are (0, 2).

As the length of *CO* is 2 units and point *C* lies on the negative *x*-axis, the *x*-coordinate of point *B* is −2.

Also, as point *C* is lies on the *x*-axis, the *y*-coordinate of point *C* is 0.

So, the coordinates of point *C* are (−2, 0).

As the length of *DO* is 2 units and point *D* lies on the negative *y*-axis, the *y*-coordinate of point *D* is −2.

Also, as point *D* lies on *y*-axis, the *x*-coordinate of point *D* is 0.

So, the coordinates of point *D *are (0, −2).

#### Page No 130:

#### Question 2:

What are the coordinates of the points A and B in the figure below?

#### Answer:

**Construction: **Draw perpendiculars from points *A* and *B*, intersecting the *x*-axis at points *C* and *D* respectively.

In Δ*ACO*:

The perpendicular distance of point *A* from *x*-axis and *y*-axis is 1 unit and

Therefore, the *x*-coordinate of point *A* is and the *y*-coordinate is 1.

Therefore, the coordinates of point *A* are (, 1).

We know that sum of angles forming a linear pair is 180°.

∴ ∠*BOD** *+ ∠*BOA** *+ ∠*AOC** *= 180°

⇒ ∠*BOD* + 90° + 30° = 180°

⇒ ∠*BOD** *+ 120° = 180°

⇒ ∠*BOD* = 180° − 120° = 60°

*BO* = *AO* = 2 (Radii of the same circle)

In Δ*BDO*:

The perpendicular distance of point *B* from *x*-axis and *y*-axis isunit and 1 unit respectively.

As point *B* lies in the second quadrant therefore, the *x*-coordinate of point *B* is −1 and the *y*-coordinate is.

Therefore, the coordinates of point *B* are (−1, ).

#### Page No 130:

#### Question 3:

With the axes of coordinates chosen along two adjacent sides of a rectangle, two opposite vertices have coordinates (0, 0) and (4, 3). What are the coordinates of the other two vertices?

#### Answer:

If the coordinate axes are chosen along the two adjacent sides of a rectangle with two opposite vertices having coordinates (0, 0) and (4, 3), then the rectangle can be drawn as follows:

As *ABCO* is a rectangle.

∴ *AB* = *OC* and *BC* = *AO*

The coordinates of point *B* are (4, 3).

Therefore, the perpendicular distance of point* B *from the *x*-axis and *y*-axis is 3 units and 4 units respectively.

Point *A* of the given rectangle lies on *y*-axis. Therefore, its *x*-coordinate is 0.

Also, the perpendicular distance of point *A* from the origin = Perpendicular distance of point* B *from the *x*-axis = 3 units.

Therefore, the *x*-coordinate of point *A* is 0 and the *y*-coordinate is 3.

So, the coordinates of point *A* are (0, 3).

Point *C *of the given rectangle lies on the *x*-axis. Therefore, its *y*-coordinate is 0.

The perpendicular distance of point *C *from the origin = Perpendicular distance of point* B *from the *y*-axis = 4 units.

Therefore, the *x*-coordinate of point *C* is 4 and the *y*-coordinate is 0.

So, the coordinates of point *C* are (4, 0).

Thus, the coordinates of the other two vertices of the given rectangle are (0, 3) and (4, 0).

#### Page No 136:

#### Question 1:

The coordinates of some pairs of points are given below. Without drawing the axes of coordinates, mark these points with the left-right, up-down positions correct. Draw rectangle with these as opposite vertices. Find the coordinates of the other two vertices and the lengths of the sides of these rectangles:

(i) (3, 5), (7, 8)

(ii) (−3, 5), (−7, 1)

(iii) (6, 2), (5, 4)

(iv) (−1, −2), (−5, −4)

#### Answer:

(i)

The opposite vertices of a rectangle are (3, 5) and (7, 8).

The rectangle drawn using these vertices is as follows:

The axes of the given rectangle are parallel to the axes of coordinates.

It can be observed that vertex *C* is on the right hand side of the rectangle. Also, *BC* is parallel to the *y*-axis and coordinates (7, 8). Therefore, the *x*-coordinate of vertex *C* is 7.

Now, *DC* is parallel to *x*-axis and one point on it has coordinates (3, 5). Therefore, the *y*-coordinate of vertex *C* is 5.

Therefore, the coordinates of vertex *C* are (7, 5).

Similarly, vertex *A* is on the top-left of the rectangle. Also, *AD* is parallel to *y*-axis and one point on it has coordinates (3, 5). Therefore, the *x*-coordinate of vertex *A* is 3.

Now, *AB* is parallel to *x*-axis and one point on it has coordinates (7, 8). Therefore, the *y*-coordinate of vertex *A* is 8.

Therefore, the coordinates of vertex *A* are (3, 8).

Thus, the coordinates of the other two vertices of the given rectangle are (7, 5) and (3, 8).

Length of the rectangle = *AB*

= |*x*-coordinate of vertex *B* − *x*-coordinate of vertex *A*| units

=|7 − 3| units

= 4 units

Breadth of the rectangle =* BC*

= |*y*-coordinate of vertex *C* − *y*-coordinate of vertex *B*| units

= |8 − 5| units

= 3 units

Thus, the lengths of the sides of the rectangle are 4 units and 3 units.

(ii)

The opposite vertices of a rectangle are (−3, 5) and (−7, 1).

The rectangle drawn using these vertices is as follows:

The axes of the given rectangle are parallel to the axes of coordinates.

It can be observed that vertex *C* is on the right hand side of the rectangle. Also, *BC* is parallel to *y*-axis and one point on it has coordinates (−3, 5). Therefore, the *x*-coordinate of vertex *C* is −3.

Now, *DC* is parallel to *x*-axis and one point on it has coordinates (−7, 1). Therefore, the *y*-coordinate of vertex *C* is 1.

Therefore, the coordinates of vertex *C* are (−3, 1).

Similarly, vertex *A* is on the top-left of the rectangle. Also, *AD* is parallel to *y*-axis and one point on it has coordinates (−7, 1). Therefore, the *x*-coordinate of vertex *A* is −7.

Now, *AB* is parallel to *x*-axis and one point on it has coordinates (−3, 5). Therefore, the *y*-coordinate of vertex *A* is 5.

Therefore, the coordinate of vertex *A* are (−7, 5).

Thus, the coordinates of the other two vertices of the given rectangle are (−3, 1) and (−7, 5).

Length of the rectangle = *AB*

= |*x*-coordinate of vertex *B* − *x*-coordinate of vertex *A*| units

= |−3 − (−7)| units

= |−3 + 7| units

= 4 units

Breadth of the rectangle =* BC*

= |*y*-coordinate of vertex *C* − *y*-coordinate of vertex *B*| units

= |1 − 5| units

= 4 units

Thus, the lengths of the sides of the rectangle are 4 units each.

(iii)

The opposite vertices of a rectangle are (6, 2) and (5, 4).

The rectangle drawn using these vertices is as follows:

The axes of the given rectangle are parallel to the axes of coordinates.

It can be observed that vertex *B* is on the top-right of the rectangle. Also, *BC* is parallel to *y*-axis and one point on it has coordinates (6, 2). Therefore, the *x*-coordinate of vertex *B* is 6.

Now, *AB* is parallel to *x*-axis and one point on it has coordinates (5, 4). Therefore, the *y*-coordinate of vertex *B* is 4.

Therefore, the coordinates of vertex *B* are (6, 4).

Similarly, vertex *D* is on the left hand side of the rectangle. Also, *AD* is parallel to *y*-axis and one point on it has coordinates (5, 4). Therefore, the *x*-coordinate of vertex *D* is 5.

Now, *DC* is parallel to *x*-axis and one point on it has coordinates (6, 2). Therefore, the *y*-coordinate of vertex *D* is 2.

Therefore, the coordinate of vertex *D *are (5, 2).

Thus, the coordinates of the other two vertices of the given rectangle are (6, 4) and (5, 2).

Length of the rectangle = *AB*

= |*x*-coordinate of vertex *B* − *x*-coordinate of vertex *A*| units

= |6 − 5| units

= 1 unit

Breadth of the rectangle =* BC*

= |*y*-coordinate of vertex *C* − *y*-coordinate of vertex *B*| units

= |2 − 4| units

= 2 units

Thus, the lengths of the sides of the rectangle are 1 unit and 2 units.

(iv)

The opposite vertices of a rectangle are (−1, −2) and (−5, −4).

The rectangle drawn using these vertices is as follows:

The axes of the given rectangle are parallel to the axes of coordinates.

It can be observed that vertex *C* is on the right hand side of the rectangle. Also, *BC* is parallel to *y*-axis and one point on it has coordinates (−1, −2). Therefore, the *x*-coordinate of vertex *C* is −1.

Now, *DC* is parallel to *x*-axis and one point on it has coordinates (−5, −4). Therefore, the *y*-coordinate of vertex *C* is −4.

Therefore, the coordinates of vertex *C* are (−1, −4).

Similarly, vertex *A* is on the top-left of the rectangle. Also, *AD* is parallel to *y*-axis and one point on it has coordinates (−5, −4). Therefore, the *x*-coordinate of vertex *A* is −5.

Now, *AB* is parallel to *x*-axis and one point on it has coordinates (−1, −2). Therefore, the *y*-coordinate of vertex *A* is −2.

Therefore, the coordinate of vertex *A* is (−5, −2).

Thus, the coordinates of the other two vertices of the given rectangle are (−1, −4) and (−5, −2).

Length of the rectangle = *AB*

= |*x*-coordinate of vertex *B* − *x*-coordinate of vertex *A*| units

= |−1 − (−5)| units

= |−1 + 5| units

= 4 units

Breadth of the rectangle =* BC*

= |*y*-coordinate of vertex *C* − *y*-coordinate of vertex *B*| units

= |−4 − (−2)| units

= |−4 + 2| units

= 2 units

Thus, the lengths of the sides of the rectangle are 4 units and 2 units.

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