The coordinates of points A and B are A(2,3).B(4,-5).M is the midpoint of vector AB.Determine the coordinates of point M and the magnitude of vector BM.
A(2, 3), M(x, y), B(4, -5).
AM = AB/2
x-2 = (4-2)/2
X = 3.
y-3 = (-5-3)/2
Y =
To find the coordinates of point M, we need to calculate the average of the x-coordinates and the average of the y-coordinates of points A and B.
Let's start with the x-coordinate of point M:
x-coordinate of M = (x-coordinate of A + x-coordinate of B) / 2
= (2 + 4) / 2
= 6 / 2
= 3
Next, we'll find the y-coordinate of point M:
y-coordinate of M = (y-coordinate of A + y-coordinate of B) / 2
= (3 + -5) / 2
= -2 / 2
= -1
Therefore, the coordinates of point M are M(3, -1).
To calculate the magnitude of vector BM, we need to find the length of vector BM. The length of a vector can be found using the distance formula, which states that the length of a vector (AB) with coordinates (x1, y1) and (x2, y2) is given by:
Length of AB = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, the coordinates of B are (4, -5) and the coordinates of M are (3, -1).
Length of BM = √((3 - 4)^2 + (-1 - (-5))^2)
= √((-1)^2 + (4)^2)
= √(1 + 16)
= √17
Therefore, the magnitude of vector BM is √17 (approximately 4.123).
To find the coordinates of point M, which is the midpoint of vector AB, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint between two points (x1,y1) and (x2,y2) can be found by taking the average of the x-coordinates and the average of the y-coordinates.
Given that point A has coordinates A(2,3) and point B has coordinates B(4,-5), we can find the coordinates of point M using the midpoint formula as follows:
x-coordinate of M = (x-coordinate of A + x-coordinate of B) / 2
y-coordinate of M = (y-coordinate of A + y-coordinate of B) / 2
Using the coordinates of A and B, we substitute the values into the formulas:
x-coordinate of M = (2 + 4) / 2 = 6 / 2 = 3
y-coordinate of M = (3 + (-5)) / 2 = -2 / 2 = -1
Therefore, the coordinates of point M are M(3, -1).
To find the magnitude of vector BM, we need to calculate the distance between points B and M. The distance formula is commonly referred to as the Pythagorean theorem in two dimensions.
The distance between two points (x1,y1) and (x2,y2) can be found using the following formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Using the coordinates of B and M, we can substitute the values into the distance formula:
Distance = √[(3 - 4)^2 + (-1 - (-5))^2]
= √[(-1)^2 + (4)^2]
= √[1 + 16]
= √17
Therefore, the magnitude of vector BM is √17.
M is the average of A and B (both x and y).
That is, M = (A+B)/2
so also |BM| = 1/2 |AB| since length is a linear function
(you can use the distance formula, and show that 1/2 factors out)