Consider the points on a line segment in the coordinate plane A(–9, –1) and B(–7, –6). What is the distance from the midpoint to the origin?

To find the distance from the midpoint of the line segment AB to the origin, we first need to determine the midpoint itself.

1. Midpoint Formula: The coordinates of the midpoint M of a segment with endpoints A(x₁, y₁) and B(x₂, y₂) is given by:

[ M = left(frac{x₁ + x₂}{2}, frac{y₁ + y₂}{2}right) ]

2. Calculate the Midpoint:

– For A(–9, –1) and B(–7, –6):

[

M = left(frac{-9 + (-7)}{2}, frac{-1 + (-6)}{2}right) = left(frac{-16}{2}, frac{-7}{2}right) = (-8, -frac{7}{2})

]

3. Distance to the Origin: The distance D from a point (x, y) to the origin (0, 0) is calculated using the distance formula:

[ D = sqrt{x^2 + y^2} ]

Substituting the coordinates of the midpoint:

[

D = sqrt{(-8)^2 + left(-frac{7}{2}right)^2} = sqrt{64 + frac{49}{4}} =

To find the distance from the midpoint of the line segment AB to the origin, we first need to calculate the midpoint.

The formula for the midpoint ( M(x, y) ) between two points ( A(x_1, y_1) ) and ( B(x_2, y_2) ) is:

[

M = left( frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2} right)

]

For points ( A(-9, -1) ) and ( B(-7, -6) ):

[

M = left( frac{-9 + (-7)}{2}, frac{-1 + (-6)}{2} right) = left( frac{-16}{2}, frac{-7}{2} right) = (-8, -frac{7}{2})

]

Next, we calculate the distance from the midpoint ( M(-8, -frac{7}{2}) ) to the origin ( O(0, 0) ) using the distance formula:

[

d = sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}

]

Where ( (x_1, y_1) ) is the midpoint and ( (x_2, y_2) ) is the origin: