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?
A 2√305
B 4√305
C √305/2
D √305/4
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?
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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: