Mathematics Part i Solutions Solutions for Class 10 Math Chapter 6 Coordinates are provided here with simple step-by-step explanations. These solutions for Coordinates are extremely popular among class 10 students for Math Coordinates Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Mathematics Part i Solutions Book of class 10 Math Chapter 6 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Mathematics Part i Solutions Solutions. All Mathematics Part i Solutions Solutions for class 10 Math are prepared by experts and are 100% accurate.
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 ΔACO:
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 ΔBDO:
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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