24. Predict the type of quadrilateral that is formed by the points of intersection of the lines

3x + y - 4 = 0, 4x - 5y + 30 = 0,

y = – 3x – 1, and - 4x + 5y + 10 = 0. Give

reasons for your prediction. Verify that your

prediction is correct by solving this problem.

Step-by-step explanation:

Given the equation of the lines 3x + y - 4 = 0, 4x - 5y + 30 = 0,  y = – 3x – 1, and - 4x + 5y + 10 = 0. For us to be able to predict the type of quadrilateral formed by this lines, we need to find the slope of each of the lines. Any two lines with the same slope will be parallel lines.

The standard form of equation of a line is expressed as y = mx+c where m is the slope of the line.

Re-writing each equation in standard format:

For the line 3x + y - 4 = 0;

3x+y = 4

y = 4-3x

y = -3x + 4

Hence the slope of the line is -3

For the line 4x - 5y + 30 = 0:

4x-5y = -30

-5y = -30-4x

-5y = -4x-30

divide through by -5

y = 4/5 x + 6

Hence the slope of this line is 4/5

For the line y = – 3x – 1:

The line is already in its standard format, hence the slope of the line is -3

For the line - 4x + 5y + 10 = 0:

-4x+5y = -10

5y = -10+4x

5y = 4x-10

divide through by 5

y = 4/5 x - 10/5

y = 4/5 x - 2

Hence the slope of this line is 4/5

From the slopes above, it can be seen that the line 4x - 5y + 30 = 0 is parallel to  - 4x + 5y + 10 = 0 and that of the line 3x + y - 4 = 0 is parallel to y = – 3x – 1 due to their similar slope.

To know the the type of quadrilateral formed by this lines, first we need to check whether two dissimilar lines are perpendicular to each other. If they are perpendicular to each other, it means the quadrilateral could be a square or a rectangle but if otherwise, it will either be a parallelogram or a rhombus.

For two lines to be perpendicular, the product of their slope must be -1 i.e m1m2 = -1

For the first two lines:

m1 = -3 and m2 = 4/5

Taking their product:

-3*4/5 = -12/5

Since the product did not give use -1, hence the quadrilateral is neither a square nor a rectangle. The quadrilateral can therefore either be a parallelogram or a rhombus since the intersecting lines of either shapes are not perpendicular to each other.