Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Functions - Linear models - gradient, distance and midpoint.
Test Yourself 1.


 

 

Given diagrams.

1. Using the points A (2, 2) and O (0, 0) in the above diagram:

(i) Draw a line from O to (2, 0) and then join the point (2, 0) to A (2, 2).

(ii) Mark the lengths of the two lines you have just drawn on the the diagram.

(iii) Using the lengths you have marked, calculate the gradient of the line AO.

(iv) Using Pythagoras' Theorem, calculate the length of AO.

(v) By averaging the x coordinates and then the y coordinates, calculate the mid-point of AO.

Answer.(iii) Gradient = 1
(iv) Length = 2√2
(v) Midpoint is (1, 1).
 

2. Using the points A (-1, 2) and B (1, -2) in the above diagram:

(i) Draw a line from A down to (-1, -2) and then join the point (1, -2) to (-1, -2).

(ii) Mark the lengths of the two lines you have just drawn on the the diagram.

(iii) Using the lengths you have marked and checking the direction of the slope, calculate the gradient of the line AB.

(iv) Using Pythagoras' Theorem, calculate the distance between A and B.

(v) By averaging the x coordinates and then the y coordinates, calculate the mid-point of AB.

Answer.(iii) Gradient = 2
(iv) Length = 2√5
(v) Midpoint is (0, 0).
Drawing your own diagram. 3. Plot the points P (-3, 5) and Q (5, -2) on a set of axes.

(i) Draw two lines to the points P and Q so that they form a right angle at R (-3, -2).

(ii) Mark the lengths of the two lines you have just drawn on the the diagram.

(iii) Using the lengths you have marked, calculate the gradient of the line PQ.

(iv) Using Pythagoras' Theorem, calculate the distance between P and Q.

(v) By averaging the x coordinates and then the y coordinates, calculate the mid-point of PQ.

Answer.(iii) Gradient = -7/8
(iv) Length = 10.6
(v) Midpoint is (1, 2/3).
  4. Plot the points M (-5, -8) and N (10, 16) on a set of axes.

By marking in the appropriate distance(s):

(i) Calculate the gradient of the line MN.

(ii) Calculate the distance between M and N.

(iii) Calculate the mid-point of MN.
  5. Plot the points A (-2, 1) and B (3, 1) on a set of axes.

By marking in the appropriate distance(s):

(i) Calculate the gradient of the line AB.

(ii) Calculate the distance between A and B.

(iii) Calculate the mid-point of AB.

Answer.(i) Gradient = 0
(ii) Length = 5
(iii) Midpoint is (0.5, 1).
  6. The points are C (-1, 3) and D (-1, -2).

(i) Calculate the gradient of the line AB.

(ii) Calculate the distance between A and B.

(iii) Calculate the mid-point of AB.

Answer.(iii) Gradient cannot be defined
as the line is vertical.
(ii) Length = 5
(v) Midpoint is (-1, 0.5).
Miscellaneous 7. A (-1, -2) and B (7, y) are two points on the Cartesian plane.

The gradient of the line AB is 2.

Find the value of y.

Answer.y = 14.
 

8. P (1, y) and Q (5, 1) are two points on the Cartesian plane.

The gradient of the line PQ is -2.5.

Find the value of y.

Answer.y = 11.
  9. The point L has coordinates (4, 3). The point M has coordinates (-6, -y).

The distance LM between the points is 5√5.

What are the two possible points for M? Explain why there are two possible answers and indicate the difference in the respective gradients for LM.

Answer.The points are (-6, -2) and (-6, 8).
  10. F (-3, 4) and G (a, b) are two points. The midpoint along the line joining these two points has coordinates (2, -4).

Find the values of a and b.

Answer.(a, b) is (7, -12).