GEOMETRY
Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
Name______________________________
M4
Period: _______ Date_________________
Lesson 12: Dividing Segments Proportionately
Learning Target

I can find midpoints of segments and points that divide segments two or more proportional or equal parts.
Opening Exercise (15 minutes)
Points 𝐴(−4,5), 𝐵(12,13) and 𝐶(12, 9) are plotted
on the coordinate grid
 What is the length of ̅̅̅̅
𝐴𝐶 ?

̅̅̅̅ ?
What is the length of 𝐵𝐶

Mark the halfway point on ̅̅̅̅
𝐴𝐶 and label it point .
What are the coordinates of point𝑃?

̅̅̅̅ and label it point
Mark the halfway point on 𝐵𝐶
𝑅. What are the coordinates of point 𝑅?

̅̅̅̅ perpendicular to 𝐴𝐶
̅̅̅̅ .
Draw a segment from 𝑃 to 𝐴𝐵
Mark the intersection point 𝑀. What are the
coordinates of 𝑀?

̅̅̅̅ perpendicular to
Draw a segment from 𝑅 to 𝐴𝐵
̅̅̅̅
𝐵𝐶 . Mark the intersection point 𝑀. What are the
coordinates of 𝑀?
̅̅̅̅.
Point 𝑴 is called the _____________________ of 𝑨𝑩
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?
The general formula for the midpoint of a segment with endpoints
(𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) using the average formula:
𝑥1 +𝑥2 𝑦1 +𝑦2
𝑀(
2
,
2
)
GEOMETRY
Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
Name______________________________
M4
Period: _______ Date_________________
Example 1. Find the midpoint of ̅̅̅̅
𝑺𝑻 given 𝑺(−𝟐, 𝟖) and 𝑻(𝟏𝟎, −𝟒). (Sketch the situation)
Example 2. Find the point that is one-quarter of the way along of ̅̅̅̅
𝑺𝑻 given 𝑺(−𝟐, 𝟖) and
𝑻(𝟏𝟎, −𝟒). (Sketch the situation)
Example 3. 𝑴(−𝟐, 𝟏𝟎) is the midpoint of segment ̅̅̅̅
𝑨𝑩. If A has coordinates (𝟒, −𝟓), what are the
coordinates of 𝑩?
7
̅̅̅̅ and point 𝑅 that lies on 𝑃𝑄
̅̅̅̅ such that point 𝑅 lies of the length of 𝑃𝑄
̅̅̅̅ from point 𝑃
Example 4 . Given 𝑃𝑄
along ̅̅̅̅
𝑃𝑄 .
Use the given information to determine the following ratios:
𝑃𝑅: 𝑃𝑄 =
𝑅𝑄: 𝑃𝑄 =
𝑃𝑅: 𝑅𝑄 =
𝑅𝑄: 𝑃𝑅 =
9
GEOMETRY
Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
Name______________________________
M4
Period: _______ Date_________________
2
Example 5. Given points 𝐴(−3,5) and 𝐵(12,15), find the coordinates of the point, 𝐶, that sits of the way
5
along the segment ̅̅̅̅
𝐴𝐵 , closer to 𝐴 than it is to 𝐵. (Sketch the situation)
Divide the segment based on a part:whole ratio
To partition (divide) the segment into smaller parts when part to whole ratio is given:
1. Multiple the difference in 𝑥-coordinates (𝑥2 − 𝑥1 ) and the difference in 𝑦–coordinates (𝑦2 − 𝑦1 ) by
𝑝𝑎𝑟𝑡
the given ratio (𝑤ℎ𝑜𝑙𝑒).
Then add those products to the original point (𝑥1 , 𝑦1 ) to find the partition point (𝑥𝑃 , 𝑦𝑃 ).
Another way to look at it: Recall point-slope form of a linear equation, 𝒚 − 𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏 ). By adding 𝒚𝟏
to both sides, you get = 𝒎(𝒙 − 𝒙𝟏 ) + 𝒚𝟏 . Finding a partition point uses a similar formula:
𝒑𝒂𝒓𝒕
𝒙𝑷 = 𝒘𝒉𝒐𝒍𝒆 (𝒙𝟐 − 𝒙𝟏 ) + 𝒙𝟏
and
𝒑𝒂𝒓𝒕
𝒚𝑷 = 𝒘𝒉𝒐𝒍𝒆 (𝒚𝟐 − 𝒚𝟏 ) + 𝒚𝟏
Division of the segment given as a part: part ratio  convert to part:whole
Example 6. Find the point on the directed segment from (−𝟑, 𝟎) to (𝟓, 𝟖) that divides it in the ratio of 𝟏: 𝟑.
(Sketch the situation)
Example 7. Find the point on the directed segment from (−4, 5) to (12, 13) that divides it into a ratio of 1: 7.
(Sketch the situation)
GEOMETRY
Name______________________________
Period: _______ Date_________________
Example 8. Given points 𝑨(𝟑, −𝟗) and 𝑩(𝟏𝟗, −𝟏), find the coordinates of point 𝑪 such that
or
M4
Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
𝑨𝑪 = 𝟕𝑪𝑩 . (Sketch the situation)
𝑪𝑩
𝑨𝑪
𝟏
= .
𝟕
Example 9. Find the coordinates of point 𝑃 along the directed line
3
segment ̅̅̅̅
𝐴𝐵 so that the ratio of 𝐴𝑃 to 𝑃𝐵 is .
2
Example 10. Given points 𝑨(𝟑, −𝟓) and 𝑩(𝟏𝟗, −𝟏), find the coordinates of point 𝑪 such that
Sketch the situation
𝑪𝑩
𝑨𝑪
𝟑
= .
𝟕
GEOMETRY
Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
Name______________________________
M4
Period: _______ Date_________________
Lesson 12: Dividing Segments Proportionately
Classwork
Finding Midpoints
1. Find the midpoint of the given line segment at right.
2. Find the midpoint between (−2, 3) and (4, 2).
̅̅̅̅ . Find the coordinates of 𝑀 when:
3. Point 𝑀 is the midpoint of segment 𝐴𝐶
1. 𝐴(2, 3) 𝑎𝑛𝑑 𝐶(6, 10)
2. 𝐴(−7, 5) 𝑎𝑛𝑑 𝐶(4, −9)
̅̅̅ has endpoints of 𝑆(−2,4) 𝑎𝑛𝑑 𝑇(−6,0). Find the midpoint of segment ̅𝑆𝑇
̅̅̅.
4. Segment ̅𝑆𝑇
5. Points 𝑋 (7, 2) 𝑎𝑛𝑑 𝑌 (5, −4) connect to make segment ̅̅̅̅
𝑋𝑌. Find the midpoint of this segment.
6. What is the midpoint of a segment whose endpoints are 𝑋(6, 4) 𝑎𝑛𝑑 𝑌 (9, −2)?
GEOMETRY
Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
Name______________________________
M4
Period: _______ Date_________________
7. A circle has a center of (2, −2). The diameter has one endpoint of (4,2). What are the coordinates of the other
endpoint?
A.
B.
C.
D.
(−4,2)
(0, −6)
(−3, −3)
(8,4)
8. A player kicks a soccer ball from a position that is 10 yards from a
sideline and 5 yards from a goal line. The ball lands at a position that is
halfway to his teammate, who is 45 yards from the same goal line and
40 yards from the same sideline. Where did the ball end up?
9. What are the coordinates of the center of a circle if the endpoints of its diameter are
𝐴(8, −4) 𝑎𝑛𝑑 𝐵(−3,2)?
10. Square LMNO is shown in the diagram below. Find the coordinates of
the midpoint of diagonal
?
̅̅̅̅ has endpoints 𝐴(3𝑥 + 5, 3𝑦) and 𝐵(𝑥 − 1, −𝑦). What are the coordinates of the
11. Line segment 𝐴𝐵
midpoint of ̅̅̅̅
𝐴𝐵 ?
1)
2)
3)
4)
GEOMETRY
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 12
M4
Name______________________________
Period: _______ Date_________________
12. Point 𝑀 is the midpoint of
are the coordinates of B?
and the coordinates of 𝑀 are
. If the coordinates of A are
1)
2)
3)
4)
Divide the segment based on a part: part ratio and part: whole ratio
𝟑
13. Given points 𝑨(𝟑, −𝟓) and 𝑩(𝟏𝟗, −𝟏), find the coordinates of point 𝑪 that sit 𝟖 of the way along
̅̅̅̅
𝑨𝑩, closer to 𝑨 than to 𝑩.
𝟓
14. Given 𝑭(𝟎, 𝟐) and 𝑮(𝟐, 𝟔). If point 𝑺 lies 𝟏𝟐 of the way along ̅̅̅̅
𝑭𝑮, closer to 𝑭 than to 𝑮, find the
coordinates of 𝑺.
15. Find the point on the directed segment from (−𝟑, −𝟐) to (𝟒, 𝟖) that divides it into a ratio of 𝟑: 𝟐.
, what
GEOMETRY
Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
Name______________________________
M4
Period: _______ Date_________________
𝑪𝑩
𝟑
16. Given points 𝑨(𝟑, −𝟓) and 𝑩(𝟏𝟗, −𝟏), find the coordinates of point 𝑪 such that 𝑨𝑪 = 𝟕.
Sketch the situation
17. What are the coordinates of the point that would divide the segment with endpoints from points
𝑷𝟏(𝟐, 𝟖) and 𝑷𝟐(𝟕, 𝟑) into two segments with the ration of 4:1?
18. Segment AB is drawn from A(𝟎, 𝟏𝟎) to 𝑩(𝟎, 𝟏𝟎) . Find point C that partitions segment AB in the
ratio 5: 2
19. ***Two runners are moving towards each other on a straight road
that is represented on the coordinate plane. One runner is moving
1.5 times faster than the other runner. If the faster runner is
currently located at (−𝟑, 𝟐) and the slower runner is currently
located at (𝟏𝟎, −𝟑), at what point will the two runners meet?