Geometry- Determine the equation of a right bisector.

Question

Determine the equation of the right bisector of the line segment E(2,6) and F(4,-2). Draw the diagram.

Answer 1

Step 1: Find the midpoint of EF. Since we are trying to find the equation of a perpendicular bisector and a bisector divides a line into two equal parts, we will use the midpoint formula to find the midpoint of EF.

x₁ + x₂ y₁ + y₂

( ---------- , ---------- )

2 2

Consider E(2, 6) to be (x₁ , y₁) and F(4, -2) to be (x₂ , y₂).

2+4 6+(-2)

( ---------- , ---------- )

2 2

6 4

( ---------- , ---------- )

2 2

(3, 2)

Step 2: Find the slope of EF using the slope formula. Consider E(2, 6) to be (x₁ , y₁) and F(4, -2) to be (x₂ , y₂).

y₂ - y₁

m = ----------

x₂ - x₁

-2-6

m = ----------

4-2

-8

m = -----------

2

m = -4

Step 3: The equation we are told to find is perpendicular to EF, and perpendicular lines have slope that are opposite reciprocal of one another. So, the opposite of -4 is 4 and the reciprocal of 4 is 1/4.

The slope of the perpendicular line is 1/4.

Step 4: Find the b value for the equation of the perpendicular line. We know it has a slope of 1/4 and goes through (3, 2). All coordinates are written in the form (x, y) so in the coordinate (3, 2), 3 is x and 2 is y. Plug these in to find the b value.

y = mx + b

2 = 1/4(3) + b

2 = 3/4 + b

-3/4 -3/4

---------------

5/4 = b

Step 5: Write the equation of the perpendicular line in y = mx+b form.

y = 1/4 x + 5/4

Answer 2

A perpendicular or right bisector goes through the midpoint and is perpendicular to the given line. Two perpendicular lines have slopes that when multiplied together = -1. This is another way of stating that the slopes are negative reciprocals of each other.

To determine the slope of the given line put the difference of the y's over the difference of the x's. Here 6 - (-2) for the y's = 8 and 2 - 4 = -2 for the x's. 8 /-2 = -4 for this slope. The new line would have a slope of 1/4.

The midpoint is found for each variable as the sum of the values for that variable over 2. Here for x: (2 + 4) / 2 = 3 and ( 6+ (-2))/2 = 2. Thus the midpoint is 3 ,2.

The equation for the line with a slope of 1/4 that goes through 3, 2 must now be found. One substitutes in the midpoint point and slope into the slope- intercept equation of a line. 2 = (1/4 ) times 3 + b. B =1 1/4.

The perpendicular bisector line equation is y = x/4 + 1 1/4.

Answer 3

The equation that will need to be used will be the point/slope equation. For a line passing through the point (x1, y1) with a slope of m, the equation is:

(y-y1) = m*(x-x1)

You are asked to find the right bisector. This will pass through the midpoint of the given line segment. The midpoint will be given ((x2+x1)/2, (y2+y1)/2). For the points given, this means the midpoint is at

((4+2)/2, (-2+6)/2) or (3,2)

Now you need to determine m. To do that, recall that for a given line of slope m, the perpendicular line has slope -1/m.

The slope of the line segment is given by (y2-y1)/(x2-x1), so the slope of the line segment is

(-2-6)/(4-2) = -4. Since the slope of the line segment is -4, the slope of the perpendicular line is -1/-4 = 1/4

Now we have the point and the slope of the right bisector. Plugging into the point/slope formula gives

(y-2)= (1/4)*(x-3). This simplifies to y-2= (1/4)x-(3/4). The final equation of the line is therefore

y=(1/4)x+(5/4)