How to solve parallel and perpendicular lines?

Question

Write the standard form of the equations of the lines that are parallel and perpendicular to y=2x-3 and pass though the point with the given coordinates.

 

1.) (4,2)

 

2.) (8,-2)

 

3.) Show that the quadrilateral WXYZ is a square if it's vertices are W (-1,3), X (3,6), Y (6,2), and Z (2,-1).

 

4.) Show that triangle ABC is  a right triangle if its vertices are A (-1,2), B (4,-3), and C (-2,1).

 

5.) Prove analytically that the line joining the midpoints of the nonparallel sides of a trapezoid I'd parallel to the bases of the trapezoid. 

Answer 1

Hi Giovanni,

 

Lots of good problems here !!

 

We'll take a look at them one by one in order.

 

Problem #1:

 

1.  We are given line (y = 2x - 3) and are asked to find a line parallel to it which passes through point (4,2).

 

2.  First, we need to make sure that the equation of the given line is in slope-intercept form:

         y = mx + b (m is slope and b is y-intercept)

 

3.  We see that it is and that the slope of the given line is 2 (the coefficient of x in the equation).

 

4.  Parallel lines have equal slopes, so we now have the general equation of a line, which is parallel to the given line (y = 2x -3) with slope = 2:

         y = 2x + b

 

5.  We can calculate b (y-intercept) by plugging the coordinates of our given point (4,2) into the general equation from previous step:

         y = 2x + b

         2 = 2(4) + b

         2 = 8 + b

         b = 2 - 8

         b = -6

         

6.  Now we can write the specific equation in slope intercept form for the line that is parallel to the given line (y = 2x - 3) and which also passes through given point (4,2)

         y = 2x - 6 (slope, m = 2; y-intercept, b = -6)

 

7.  Perpendicular lines have slopes which are the negative reciprocals of each other, so the general equation of a line, which is perpendicular to the given line (y = 2x -3) with slope = -1/2 (negative reciprocal of 2):
         y = -(1/2)x + b

8.  We can calculate b (y-intercept) by plugging the coordinates of our given point (4,2) into the general equation from previous step:
         y = -(1/2)x + b
         2 = -(1/2)(4) + b
         2 = -2 + b

         b = 2 + 2
         b = 4

9.  Now we can write the specific equation in slope intercept form for the line that is perpendicular to the given line (y = 2x - 3) and which also passes through given point (4,2):
         y = -(1/2)x + 4
                  (slope, m = -(1/2); y-intercept, b = 4)

 

Problem #2:

 

1.  You can follow the same procedure we used for Problem #1.

 

2.  The only difference is the given point (8,-2).  To calculate b (y-intercept), you will need to plug in 8 for x and plug in -2 for y into the general equation for each line.

 

Problem #3:

 

1.  We are given quadrilateral WXYZ and are asked to show that it is a square.

 

2.  Below is a clickable link to the figure of given quadrilateral WXYZ with its given coordinates for reference:

 

http://www.snag.gy/0WQhf.jpg

  

3.  To show that this figure is a square we will need to show two things:  the lengths of all the sides are equal and all four vertex angles are right angles.

 

4.  To show that the lengths of all the sides are equal we can use the distance formula to calculate the length of each side:

         d = sqrt((x₂ - x₁)² + (y₂ - y₁)²)

         WX = sqrt((3 - (-1))² + (6 - 3)²) = 5

         XY = sqrt((6 - 3)² + (2 - 6)²) = 5

         YZ = sqrt((2 - 6)² + (-1 - 2)²) = 5

         ZW = sqrt((2 - (-1))² + (-1 - 3)²) = 5

 

      All sides have equal length of 5.

 

5.  To show that all four vertex angles are right angles, we need to use the slope formula to show that the lines forming each vertex angle are perpendicular lines (lines with negative reciprocal slopes):

         m (slope) = (y₂ - y₁) / (x₂ - x₁)

         m (WX) = (6 - 3) / (3 - (-1)) = 3/4

         m (XY) = (2 - 6) / (6 - 3) = -4/3

         m (YZ) = (-1 - 2) / (2 - 6) = 3/4

         m (ZW) = (-1 - 3) / (2 - (-1)) = -4/3

 

     All four vertex angles are right angles:

         /_ W is a right angle (ZW _|_ WX)

         /_ X is a right angle (WX _|_ XY)

         /_ Y is a right angle (XY _|_ YZ)

         /_ Z is a right angle (YZ _|_ ZW)

 

Problem #4:

 

1.  You can follow the same procedure we used for Problem #3 (step 4) to cacuclate the lengths of the three sides of the triangle.

2.  You can then use the Pythagorean Theorem (for right triangles) using sides (a, b, c) to show that the three sides form a right triangle:

     Pythagorean Theorem:  a² + b² = c²

 

Problem #5: 

 

1.  Below is a clickable link to to the figure of a trapezoid (ABCD) with its given coordinates for reference:

http://www.snag.gy/j8zQh.jpg

 

2.  We can calculate midpoints E and F using the midpoint formula:

         midpoint = ((x₁ + x₂)/2,(y₁ + y₂)/2)

         midpoint E = ((6 + 1)/2,(10 + 4)/2) = (3.5,7)

         midpoint F = ((14 + 16)/2,(10 + 4)/2) = (15,7)

 

3.  You can then use the slope formula (see step 5 of Problem 3) to calculate the slopes of all three lines (base AB, base CD, mid segment EF) to show that all the all three lines are || (|| lines have equal slopes).

 

These problems provide a good review of coordinate geometry.  Thanks for submitting them and glad to be of help.

 

God bless, Jordan.