Find the midpoint of a line segment starting from the origin and ending at $(-8,0)$.
A) $(-4,0)$
B) $(0,-4)$
C) $(0,0)$
D) $(-16,0)$
Answer (1)
Identify the endpoints of the line segment: ( 0 , 0 ) and ( − 8 , 0 ) .
Apply the midpoint formula: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) .
Substitute the coordinates into the formula: M = ( 2 0 + ( − 8 ) , 2 0 + 0 ) .
Simplify to find the midpoint: ( − 4 , 0 ) .
Explanation
Problem Analysis We are given a line segment that starts at the origin ( 0 , 0 ) and ends at the point ( − 8 , 0 ) . Our goal is to find the midpoint of this line segment.
Midpoint Formula The midpoint formula is given by: M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the endpoints of the line segment.
Applying the Formula In our case, ( x 1 , y 1 ) = ( 0 , 0 ) and ( x 2 , y 2 ) = ( − 8 , 0 ) . Substituting these values into the midpoint formula, we get: M = ( 2 0 + ( − 8 ) , 2 0 + 0 ) M = ( 2 − 8 , 2 0 ) M = ( − 4 , 0 ) So, the midpoint of the line segment is ( − 4 , 0 ) .
Final Answer The midpoint of the line segment starting from the origin and ending at ( − 8 , 0 ) is ( − 4 , 0 ) . Therefore, the correct answer is A) ( − 4 , 0 ) .
Examples
In architecture, finding the midpoint of a structural beam is crucial for evenly distributing weight and ensuring stability. For instance, if a beam spans from the origin to a point 8 meters away on the x-axis, determining the midpoint helps engineers place support columns accurately, preventing structural failure. This ensures the load is balanced, enhancing the building's overall safety and longevity.
