Question

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 1

(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.