Suppose points A, B, and C have the coordinates given below. Find y such that C is on the perpendicular bisector of segment AB.

A(−1, 4), B(7, −2), C(9, y)

wrong

Well, finding the y-coordinate of point C such that it lies on the perpendicular bisector of segment AB involves a bit of math. But worry not, I'm here to clown around and make it more entertaining for you!

So, let's think about this logically. The perpendicular bisector of a line segment cuts it into two equal halves, right? That means the distance from each endpoint to the bisector is the same.

We can start by finding the midpoint of segment AB. The x-coordinate of the midpoint should be the average of the x-coordinates of A and B, and the same goes for the y-coordinate. So, let's calculate:

Midpoint = ((-1 + 7) / 2, (4 + -2) / 2)
= (3, 1)

Now, we need to find the slope of segment AB. Using the formula (y₂ - y₁) / (x₂ - x₁), we get:

Slope of AB = (-2 - 4) / (7 - (-1))
= (-6) / 8
= -3/4

Remember, the slope of a line perpendicular to AB would be the negative reciprocal of its slope. So, the perpendicular slope is 4/3.

Since the perpendicular bisector of AB runs through the midpoint (3, 1), we can use the point-slope form of a line to find its equation:

(y - 1) = (4/3)(x - 3)

Now, to find the y-coordinate of point C, we substitute the x-coordinate 9 into the equation:

(y - 1) = (4/3)(9 - 3)
(y - 1) = (4/3)(6)
(y - 1) = 8

By solving the equation, we get:

y = 9

And voila! The y-coordinate of point C, so that it lies on the perpendicular bisector of segment AB, is 9.

I hope I didn't clown around too much with all the math. But hey, at least we found our answer!

To find the value of y such that point C is on the perpendicular bisector of segment AB, you need to follow these steps:

1. Determine the midpoint of segment AB by finding the average of the x-coordinates and the y-coordinates of points A and B.

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

Substituting the values of A and B:

Midpoint = ((-1 + 7) / 2, (4 + -2) / 2)
= (6 / 2, 2 / 2)
= (3, 1)

2. Calculate the slope of segment AB using the formula:

Slope = (y₂ - y₁) / (x₂ - x₁)

Substituting the values of A and B:

Slope = (-2 - 4) / (7 - -1)
= (-6) / 8
= -3/4

3. Determine the negative reciprocal of the slope of segment AB to find the slope of the perpendicular bisector. The negative reciprocal is found by flipping the fraction and changing the sign:

Negative Reciprocal = -1 / Slope

Negative Reciprocal = -1 / (-3/4)
= 4/3

4. Now that you have the slope of the perpendicular bisector and the midpoint, you can use the point-slope form of a linear equation to find the equation of the perpendicular bisector.

Point-Slope Form: (y - y₁) = m(x - x₁), where (x₁, y₁) is a point on the line, and m is the slope.

Substituting the midpoint (x₁, y₁) = (3, 1) and the slope m = 4/3:

(y - 1) = (4/3)(x - 3)

5. Finally, substitute the x-coordinate of point C into the equation of the perpendicular bisector and solve for y.

(y - 1) = (4/3)(9 - 3)
(y - 1) = (4/3)(6)
(y - 1) = 8

Solving for y:

y = 8 + 1
= 9

Therefore, for point C(9, y) to be on the perpendicular bisector of segment AB, y must be equal to 9.

AB has slope -6/8 = -3/4 and length 10.

So, we want a line with slope 4/3, passing through the midpoint (A+B)/2 = (3,1)

When x = 9, it's 6 units to the right of point A. So, y will be 4/3*6=8 units above point A, at 12. So, C=(9,12)