Question

Consider the points A = (-1, 5) and B = (x, y). given the midpoint M = (3, -2), find the values of x and y. Show your work in two different ways. Then, find the length of AB as well.

Answer 1

The values of x and y for point B are 7 and -9, respectively, found by setting up equations using the midpoint formula. The length of the segment AB is approximately 16.12 units, calculated using the distance formula.

Step-by-step explanation:

To find the values of x and y for point B given midpoint M, we can use the midpoint formula. The midpoint formula for two points A(x1, y1) and B(x2, y2) with midpoint M(xm, ym) is given by:

xm = (x1 + x2) / 2, ym = (y1 + y2) / 2.

So for point A = (-1, 5) and midpoint M = (3, -2), we can set up two equations:

3 = (-1 + x) / 2, -2 = (5 + y) / 2.

We solve these equations to find x = 7 and y = -9.

Now, to find the length of AB, we can use the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2).

Using point A = (-1, 5) and point B = (7, -9), the length of AB is √((7 - (-1))^2 + (-9 - 5)^2), which simplifies to √(8^2 + (-14)^2) = √(64 + 196) = √260 ≈ 16.12 units.