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mathematics UK Year 10 Mathematics (KS4)
 
Solved problems for the chapter:
 

Parallel and perpendicular lines

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Equation of a straight line for KS4
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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:parallel_and_perpendicular_lines/5330/


Example 1.

Calculate the point of the intersection of the lines:




    
Example 2.

For what value of b are the lines and parallel?

    
3.

The edges of the cuboid ABCDEFGH define the following geometric elements:


  • plane ,

  • plane ,

  • the line p,

  • the line q.


The following relationships apply between the elements:


  • The plane passes through the points C, D and H.

  • The plane passes through the points A and E and is perpendicular to the plane .

  • The line p represents the intersection of the planes and .

  • The line q lies on the plane and intersects the line p at the point D at a right angle.


Substitute the above four geometric elements correctly in the cuboid and write down the relationships between them with symbols.

    
4.

Write the relationships between the geometric elements in the diagram below:

  • point H,

  • the line p,

  • planes and

  • planes .


    
5.

In the room we have the following elements, as shown in the diagram below:




Check the correctness of the following statements:
















    
6.

The edges of the cuboid ABCDEFGH define the following geometric elements:

  • the line q,

  • planes and

  • planes .


The following relationships apply between the elements:

  • The plane passes through points A, C and D.

  • The plane has points A and D in common with the plane , and also passes through the point E.

  • The line q is perpendicular to the plane and passes through the point B.


Substitute the above three geometric elements correctly in the cuboid and write down the relationships between them with symbols.

    
7.

Write the relationships between the geometric elements in the diagram below:

  • the line p,

  • the line q,

  • the line r and

  • planes .


    
8.

In the room we have the following elements, as shown in the diagram below:




Check the correctness of the following statements:















    

9.

The edges of the cuboid ABCDEFGH define the following geometric elements:

  • plane ,

  • the line p,

  • the line q and

  • the line r.


The relationships between the elements are:

  • The plane passes through points A, D and H.

  • The line p passes through the point D and is perpendicular to the plane .

  • The line r passes through the point H and is perpendicular to the line p.

  • Line q passes through point E and is perpendicular to line r.


Substitute the above four geometric elements correctly in to the cuboid and write the relationships between them using symbols.

    
10.

Write the relationships between the geometric elements in the diagram below:

  • point A,

  • the line p,

  • the line q and

  • line r


    
11.

In the room we have the following elements, as shown in the diagram:




Check the correctness of the following statements:














    
12.

The edges of the cuboid ABCDEFGH define the following geometric elements:

  • the line p,

  • the line q and

  • the line r.


The relationships between the elements are:

  • The line q passes through the points A and D.

  • Line p passes through point B and is perpendicular to line q.

  • Line r is parallel to line p and intersects line q at point D.


Substitute the above three geometric elements correctly in to the cuboid and write the relationships between them using symbols.

    
13.

We have the following items in the room as shown in the diagram. Line p passes through points A and C, and line q passes through B and H.




Check the correctness of the following statements:















    
14.
Calculate the angle between the lines:



    
15.

The lines are given






Draw the lines in the coordinate system and describe their relative position.

    
16.

A straight line with the equation is given:




  • Write down the coordinates of the intersections of the given line with the coordinate axes and draw it in the coordinate system.


  • Write down the equation of the line that is parallel to the given line and passes through the point A(0,3).


  • Calculate and record the intersection of the given line with the line:



    
17.

Draw the graph of the equation below:




Write down the coordinates of the intersections of the line with the coordinate axes.


    
18.

Write the equation of the line passing through the point A (1,0) and parallel to the line in the graph below.


    

19.

Determine the value of a such that the line is


  • perpendicular


  • parallel


to the line

   
20.

What angle does the line makes with the ordinate axis?

    
21.

Given the lines 2 x + 5 y - 2 = 0 and y = 2 x + 2.


  • Calculate the intersection between the lines.


  • Draw both lines in the same coordinate system.


  • Calculate the angle between the lines.

    
22.

Find the equation of a line that has an angle of inclination and passes through the point of intersection of the lines




and


    
23.

Calculate the angle between the lines and .

    
24.

Calculate the angle between the lines and .

    
25.

Find the angle of slope of the line:


    
26.

Find the angle of slope of the line:


    
27.

The line p in the figure passes through the points A and B.




  • Obtain the equation of the line.


  • Calculate the size of the acute angle enclosed by the straight line and the abscissa axis. Round the result to a hundredth of a degree.


    
28.
Calculate the angle of slope of the line:

    

29.
Calculate the angle of slope of the line:

    
30.
Calculate the angle between the lines:



    
31.
Calculate the angle of slope of the line:

    
32.

A line with the equation is given. Calculate the intersections of the line with the coordinate axes and draw the line in the coordinate system.

    
33.

In the room we have the following elements, as shown in the diagram:




Check the correctness of the following statements:













    
34.

Write the relationship between the geometric elements in the diagram:


  • line p,

  • line q,

  • plane and

  • plane .


    
35.

In the room we have the following elements, as shown in the diagram:




Check the correctness of the following statements:













    
36.

With symbols, write down all the relationships between the coloured geometric elements in the diagram below:


  • point B,

  • line p,

  • line q and

  • plane .


    
37.

The edges of the cube ABCDEFGH define the following geometric elements:

  • planes ,

  • the line p,

  • the line q.


The following relationships apply between the elements:

  • The plane passes through points B, C and D.

  • Lines p and q lie on the plane and intersect at point A.


Inscribe the above three geometric elements correctly in the square and write down the relationships between them with symbols.