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Introduction The use of the coordinate plane can be helpful with many real-world applications. Scenarios can be translated into equations, equations can be graphed, points can be found, and distances between points can be calculated; but what if you need to find a point on a line that is halfway between two points? Can this be easily done? : Midpoints and Other Points on Line Segments

Key Concepts Lines continue infinitely in both directions. Their length cannot be measured. A line segment is a part of a line that is noted by two endpoints, (x 1, y 1 ) and (x 2, y 2 ). The length of a line segment can be found using the distance formula,. The midpoint of a line segment is the point on the segment that divides it into two equal parts : Midpoints and Other Points on Line Segments

Key Concepts, continued Finding the midpoint of a line segment is like finding the average of the two endpoints. The midpoint formula is used to find the midpoint of a line segment. The formula is. You can prove that the midpoint is halfway between the endpoints by calculating the distance from each endpoint to the midpoint. It is often helpful to plot the segment on a coordinate plane. 3 DELETE THIS WHILE CHECKING PICKUP: I moved the bottom bullet on this slide from after the table on the next page so things would fit better 6.2.1: Midpoints and Other Points on Line Segments

Key Concepts, continued Other points, such as a point that is one-fourth the distance from one endpoint of a segment, can be calculated in a similar way. 4 Finding the Midpoint of a Line Segment 1.Determine the endpoints of the line segment, (x 1, y 1 ) and (x 2, y 2 ). 2.Substitute the values of (x 1, y 1 ) and (x 2, y 2 ) into the midpoint formula:. 3.Simplify : Midpoints and Other Points on Line Segments

Key Concepts, continued 5 Finding the Point on a Line Segment with Any Given Ratio 1.Determine the endpoints of the line segment, (x 1, y 1 ) and (x 2, y 2 ). 2.Calculate the difference between the x-values: |x 2 – x 1 |. 3.Multiply the difference by the given ratio,. 4.Add the product to or subtract it from the x-value of the endpoint. This is the x-value of the point with the given ratio. 5.Calculate the difference between the y-values: |y 2 – y 1 |. 6.Multiply the difference by the given ratio,. 7.Add the product to or subtract it from the y-value of the endpoint. This is the y-value of the point with the given ratio : Midpoints and Other Points on Line Segments

Common Errors/Misconceptions misidentifying x 1, x 2 and y 1, y 2 attempting to use the distance formula to locate the midpoint attempting to use the midpoint formula to find points that split the segment in a ratio other than using the wrong endpoint to locate the point given a ratio incorrectly adding or subtracting units from the identified endpoint : Midpoints and Other Points on Line Segments

Guided Practice Example 1 Calculate the midpoint of the line segment with endpoints (–2, 1) and (4, 10) : Midpoints and Other Points on Line Segments

Guided Practice: Example 1, continued 1.Determine the endpoints of the line segment. The endpoints of the segment are (–2, 1) and (4, 10) : Midpoints and Other Points on Line Segments

Guided Practice: Example 1, continued 2.Substitute the values of (x 1, y 1 ) and (x 2, y 2 ) into the midpoint formula. Midpoint formula Substitute (–2, 1) and (4, 10) : Midpoints and Other Points on Line Segments

Guided Practice: Example 1, continued 3.Simplify. (1, 5.5) : Midpoints and Other Points on Line Segments

Guided Practice: Example 1, continued 11 ✔ The midpoint of the segment with endpoints (–2, 1) and (4, 10) is (1, 5.5) : Midpoints and Other Points on Line Segments

Guided Practice: Example 1, continued : Midpoints and Other Points on Line Segments

Guided Practice Example 2 Show mathematically that (1, 5.5) is the midpoint of the line segment with endpoints (–2, 1) and (4, 10) : Midpoints and Other Points on Line Segments

Guided Practice: Example 2, continued 1.Calculate the distance between the endpoint (–2, 1) and the midpoint (1, 5.5). Use the distance formula. Distance formula Substitute (–2, 1) and (1, 5.5). Simplify as needed : Midpoints and Other Points on Line Segments

Guided Practice: Example 2, continued The distance between (–2, 1) and (1, 5.5) is units : Midpoints and Other Points on Line Segments

Guided Practice: Example 2, continued 2.Calculate the distance between the endpoint (4, 10) and the midpoint (1, 5.5). Use the distance formula. Distance formula Substitute (4, 10) and (1, 5.5). Simplify as needed : Midpoints and Other Points on Line Segments

Guided Practice: Example 2, continued The distance between (4, 10) and (1, 5.5) is units. The distance from each endpoint to the midpoint is the same, units, proving that (1, 5.5) is the midpoint of the line segment : Midpoints and Other Points on Line Segments

Guided Practice: Example 2, continued 18 ✔ 6.2.1: Midpoints and Other Points on Line Segments

Guided Practice: Example 2, continued : Midpoints and Other Points on Line Segments