Write the equation of a perpendicular bisector of the segment joining the points: (7,2) and (3,0).

Okay, so I don't even know what a bisector is. Is there a formula for this or something? If some could someone tell me it. Or tell me the process.

perpendicular bisector is a line that is perpendicular to a line segment (or line joining two points) and passes through the mid-point of the line segment.

For the given points A(7,2) and B(3,0),
the mid point D is given by:
D((7+3)/2, (2+0)/2)
=D(5,1)

The slope of line segment AB is
m=(3-7)/(0-2)
=2
The slope of the perpendicular bisector
= -1/m = -(1/2)
Line passing through D and perpendicular to segment AB is
y-5=-(1/2)(x-1)
Simplify and verify.

Wait wouldn't that last part be y-1=1/2(x-5)

because five is the midpoint of the x's

You're right!

Thanks for pointing it out.

Yes, there is a formula and a process to find the equation of a perpendicular bisector. Let's first understand what a perpendicular bisector is. A perpendicular bisector is a line that intersects another line segment at its midpoint and forms a right angle (90 degrees) with it.

To find the equation of a perpendicular bisector, follow these steps:

Step 1: Find the midpoint of the line segment.
The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) can be found using the midpoint formula:

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

In this case, the endpoints are (7, 2) and (3, 0). Let's calculate the midpoint:
Midpoint = ((7 + 3) / 2, (2 + 0) / 2)
= (10/2, 2/2)
= (5, 1)

So, the midpoint of the line segment joining (7, 2) and (3, 0) is (5, 1).

Step 2: Calculate the slope of the line segment.
The slope of a line segment with endpoints (x₁, y₁) and (x₂, y₂) can be found using the slope formula:

Slope = (y₂ - y₁) / (x₂ - x₁)

In this case, the endpoints are (7, 2) and (3, 0). Let's calculate the slope:
Slope = (0 - 2) / (3 - 7)
= -2 / -4
= 1/2

So, the slope of the line segment joining (7, 2) and (3, 0) is 1/2.

Step 3: Find the negative reciprocal of the slope.
The negative reciprocal of a slope is obtained by flipping the fraction and changing its sign. In this case, the slope is 1/2, so the negative reciprocal is -2/1, which is -2.

Step 4: Use the midpoint and the negative reciprocal of the slope to find the equation of the perpendicular bisector.
The equation of a line in point-slope form is:

y - y₁ = m(x - x₁)

Where (x₁, y₁) is a point on the line and m is the slope. Substitute the values of the midpoint (5, 1) and the negative reciprocal of the slope (-2) into the equation:

y - 1 = -2(x - 5)

Now, simplify the equation:

y - 1 = -2x + 10

Finally, rearrange the equation in slope-intercept form (y = mx + b) to get the equation of the perpendicular bisector:

y = -2x + 11

So, the equation of the perpendicular bisector of the line segment joining (7, 2) and (3, 0) is y = -2x + 11.