CC Geometry H Aim #10: How do we find midpoints of segments and points that divide segments into 3, 4, or more proportional, equal parts? Do Now: a) Plot the points A(-4,5) and B(12,13) and draw AB.
b) Draw a slope Δ and label the right angle C. C B
c) The length of AC is __. The length of BC is __.
d) Mark the halfway point on AC and label it pt P. What are the coordinates of P? ______A
e) Mark the halfway point on BC and label it pt R. What are the coordinates of R? ______
f) Draw a segment from P to AB perpendicular to AC. Mark the intersection point M. What are the coordinates of M?______
g) Draw a segment from R to AB perpendicular to BC. What do you notice?
h) Point M is called the ______of AB.
i) Look at the coordinates of the endpoints and the midpoint. Can you describe how to find the coordinates of the midpoint knowing the endpoints algebraically?
Traditional Midpoint Formula: ( , )
Let's discover another way to find the midpoint. Analyze this formula which finds the same midpoint we found above:
Describe the formula in words. Starting with pt A(-4,5), find the horizontal distance between A and B, cut in 1/2, then add to the x-coordinate of pt A. Then find the vertical distance between A and B, cut in 1/2, and add to the y-coordinate of pt. A.
How would the formula change if we started at endpoint B instead of A?
Now write a general formula for the midpoint of a segment with endpoints A(x1,y1)
and B(x2,y2) starting with endpoint A: Why do you think both formulas have in them?
How would our formula change if we wanted to find the point that is one-quarter of the way along AB, closer to A than B? Find that point on the graph. ______
Now, write the formula using A to get this B point: (12,13)
A (4,5)
You may be asked the SAME question like this:
Find the point on the directed segment from (−4, 5) to (12, 13) that divides the segment so that the lengths of the two smaller segments are in a ratio of 1:3. Explain how this is the same question.
Now, find the coordinates of the point that sits of the way along AB, closer to A than B by using a formula.
Another way to ask the same question: Find the point on the directed line segment from (-4,5) to (12,13) that divides the segment into a ratio of ______.
Now, find the point on the directed line segment from (-4,5) to (12,13) that divides the segment into a ratio of 1:15 by using a formula. Exercises: 1) a. Given points A(-4,5) and B(12,13), find the coordinates of point C that sits of the way along AB, closer to A than B.
b. Can you use what you know about the slope to verify that point C lies on segment AB?
c. Given points A(-4,5) and B(12,13), find the coordinates of point D that sits of the way along AB, closer to B than A. Use the formula 2 ways to arrive at your answer.
d. Is there any possibility of points C and D coinciding? When would this happen?
e. Another way to ask part (a) above would be to ask the coordinates of the point C that partitions the directed line segment in the ratio ______.
2) Given PQ and point R that lies on PQ such that point R lies of the length of PQ from point P along PQ. a. Is point R closer to P or closer to Q, and how do you know? Make a sketch.
b. Determine the following ratios:
I. PR: PQ ______II. RQ:PQ ______III. PR:RQ ______IV: RQ: PR ______c. If the coordinates of point P are (0,0) and the coordinates of point R are (14,21), what are the coordinates of point Q? Name ______CC Geometry H Date ______HW #10
1. Find the midpoint of the segments with the following endpoints: a. (-5, 3) and (-1, -7) b. (-6, 4) and (8, 2)
2. The midpoint of LN is the point M(-2, 2). If the coordinates of point N is (2, 8), find the coordinates of point L.
3. a. Find the coordinates of the point that sits of the length of AB from point A along AB if the points are A(-7,-2) and B(14,12).
b. The same question can be asked: Find the point on the directed line segment from (-7,-2) to (14,12) that divides the segment into a ratio of ______.
4. What are the coordinates of the point on the directed line segment from A(2,-3) to B(5,13) that partitions the segment in the ratio 3:1? [Use of the grid is optional]. 5. Find the point on the directed segment from (-3,-2) to (4,8) that divides it into a ratio of 3:2.
6. What are the coordinates of the point on the directed line segment from M(-6,-3) to N(8,2) that partitions the segment into a ratio of 3 to 2?
7. The endpoints of XYZ are X(1,3) and Z(22, 18). Determine and state the coordinates of point Y, if XY: YZ = 2:1.
Review 1) As shown on the set of axes below, ∆GHS has vertices G(3,1), H(5,3) and S(1,4). Graph and state the coordinates of ∆G”H”S”, the image of ∆GHS after the
transformations: T-3,1(D(0,0),2(∆GHS))