In Exercises 19–30, find the midpoint of each line segment with the given endpoints. (√50, −6) and (√2, 6)

Verified step by step guidance

1. Step 1: Recall the formula for finding the midpoint of a line segment. The midpoint formula is: $M\; =\; (\; (x₁\; +\; x₂)/2\; ,\; (y₁\; +\; y₂)/2\; )$, where (x₁, y₁) and (x₂, y₂) are the coordinates of the endpoints.
2. Step 2: Identify the given endpoints of the line segment. Here, the endpoints are (√50, −6) and (√2, 6). Assign these values as follows: x₁ = √50, y₁ = −6, x₂ = √2, and y₂ = 6.
3. Step 3: Substitute the values of x₁ and x₂ into the formula for the x-coordinate of the midpoint: $x\_m\; =\; (\surd 50\; +\; \surd 2)/2$. This represents the average of the x-coordinates.
4. Step 4: Substitute the values of y₁ and y₂ into the formula for the y-coordinate of the midpoint: $y\_m\; =\; (-6\; +\; 6)/2$. This represents the average of the y-coordinates.
5. Step 5: Combine the results from Step 3 and Step 4 to express the midpoint as a coordinate pair: $M\; =\; (x\_m,\; y\_m)$. Simplify each component to find the final midpoint.

Key Concepts

Midpoint Formula

The midpoint of a line segment in a coordinate plane is calculated using the midpoint formula, which is given by M = ((x1 + x2)/2, (y1 + y2)/2). This formula averages the x-coordinates and the y-coordinates of the endpoints to find the point that is exactly halfway between them.

Coordinate System

A coordinate system is a two-dimensional plane defined by an x-axis (horizontal) and a y-axis (vertical). Each point in this system is represented by an ordered pair (x, y), where 'x' indicates the horizontal position and 'y' indicates the vertical position. Understanding this system is crucial for accurately plotting points and calculating distances or midpoints.

Square Root Values

Square roots are values that, when multiplied by themselves, yield the original number. In this context, √50 and √2 are used as coordinates. Recognizing how to simplify square roots and their approximate decimal values can aid in visualizing points on the coordinate plane and performing calculations involving these values.