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1.1 Line Segments, Distance and Midpoint 1.1d – Partitioning a Segment CC Standard G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. At the end of this lesson, you should be able to answer the following question: How do you find the point on a directed line segment that partitions the segment in a given ratio?

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Point P divides in the ratio 3 to 1. 1.What does this mean?

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= -1 = -7 Slope = m = =

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m = -4 5

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Slope Formula The slope of a line through the points (x 1, y 1 ) and (x 2, y 2 ) is as follows: y2 – y1y2 – y1x2 – x1x2 – x1y2 – y1y2 – y1x2 – x1x2 – x1 m =

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Ex. Find the slope of the line that passes through (–2, –3) and (4, 6). Let (x 1, y 1 ) be (–2, –3) and (x 2, y 2 ) be (4, 6). 6 – (–3) 4 – (–2) Substitute 6 for y 2, –3 for y 1, 4 for x 2, and –2 for x 1. = y 2 – y 1 x 2 – x = 3 2 = The slope of the line that passes through (–2, –3) and (4, 6) is. 3 2 *** Always reduce your fractions****

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Find the coordinate of point P that lies along the directed line segment from A(3, 4) to B(6, 10) and partitions the segment in the ratio of 3 to 2. Directed Line Segment: Tells the direction in which from which point to start and end. In this case, from Point A to Point B So point A must be labeled A and B What does that tell you about the distance AP and PB in relation to AB? Example 1 :

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Partitioning a segment 1.Label your points and Note: since it is a directed segment, order does matter. 3. Find the rise and run for the segment (order does matter) Rise: Run: 4. To find the partitioning point: x – coordinate: x 1 -coordinate + run(% in fraction form) y - coordinate: y 1 -coordinate + rise(% in fraction form) 2. Convert the ratio into a percent (keep as a fraction) a:b Percent ratio (%)

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Find the coordinate of point P that lies along the directed line segment from A(3, 4) to B(6, 10) and partitions the segment in the ratio of 3 to Convert the ratio into a percent 3. Find the rise and run for AB 4. The slope of AP must be the same as AB. So, to find the coordinates of P, add 60% of the run to the x -coordinate of A and add 60% of the rise to the y -coordinate of A How can you use the distance formula to check that P partitions in the ratio of 3 to 2? 1. Label your points. (note: Since it is a directed segment, order does matter, so A( x 1, y 1 ) and B( x 2, y 2 ) ).

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Example 2: Find the coordinates of the point P that lies along the directed segment from A(1, 1) to B(7, 3) and partitions the segment in the ratio of 1 to 4

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Example 3: Find the coordinates of point P that lies along the directed line segment from M to N and partitions the segment in the ratio of 3 to 2

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Essential question: How do you find the point on a directed line segment that partitions the segment in a given ratio?

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Class assignment: Work sheet 1.1d – Partitioning a segment

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