RATIOS - MR. ELKINS' GEOMETRY · Ratios can be shown in different ways: Use a “:” to separate the values: _ Use the word “to” to separate the values: _ PARTITIONING

© Lynn McNamara 2017

Name: Date:

Topic: Class:

Main Ideas/Questions Notes/Examples

RATIOS

A ratio compares two different quantities. It says how much of one thing there is compared to another thing.

Ratios can be shown in different ways: Use a “:” to separate the values: __________ Use the word “to” to separate the values: __________

PARTITIONING a Line Segment

To partition a line segment means to divide the line segment in the given ratio. The two numbers in the ratio must add together to equal the total number of pieces.

Label the point, P, that partitions the line

segment 𝐴𝐵̅̅ ̅̅ into a ratio of 2:3.


segment 𝐴𝐵̅̅ ̅̅ into a ratio of 1 to 4.


segment 𝐴𝐵̅̅ ̅̅ into a ratio of 3 to 2.


segment 𝐴𝐵̅̅ ̅̅ into a ratio of 4:1.

Label the point, P, that partitions the line segment 𝐵𝐴̅̅ ̅̅ into a ratio of 2:3.

Label the point, P, that partitions the line segment 𝐵𝐴̅̅ ̅̅ into a ratio of 4 to 1.

PARTITIONING a Line Segment

on the

COORDINATE

PLANE

To find the point P on a line segment between two given points that partitions the segment in a given ratio a:b, you can use the following formula:

Point P = (𝑥1 +𝑎

𝑎+𝑏(𝑥2 − 𝑥1), 𝑦1 +

𝑎

𝑎+𝑏(𝑦2 − 𝑦1) )


Point P = (𝑥1 +𝑎

𝑎+𝑏(𝑥2 − 𝑥1), 𝑦1 +

𝑎

𝑎+𝑏(𝑦2 − 𝑦1) )

Example 1:

Given the points A(3, 4) and B (6, 10), find the

coordinates of the point P on directed line segment 𝐴𝐵̅̅ ̅̅ that partitions 𝐴𝐵̅̅ ̅̅ in the ratio 2:1.

Example 2: Given the points A(-4, -3) and B (4, 1), find the

coordinates of the point P on directed line segment 𝐴𝐵̅̅ ̅̅ that partitions 𝐴𝐵̅̅ ̅̅ in the ratio 1:3.

Example 3: Given the points A(-3, -2) and B (6, 1), find the

coordinates of the point P on directed line segment 𝐵𝐴̅̅ ̅̅ that partitions 𝐵𝐴̅̅ ̅̅ in the ratio 1:2.

Example 4: Given the points A(-4, 7) and B (8, 3), find the coordinates

of the point P on directed line segment 𝐴𝐵̅̅ ̅̅ that partitions 𝐴𝐵̅̅ ̅̅ in the ratio 3:1.


Partitioning a Line Segment Pyramid Puzzle

Objective: Students will practice partitioning line segments and finding midpoints using this pyramid puzzle. This activity was written for a high school level geometry class. Students can work the problems individually or they can work in pairs.

Activity Directions:

1. Print templates and problems for each student or each pair of students. Personally, I like students to work in pairs and my copies are limited, so I copy a class set of the template and problems on brightly colored card stock and use them throughout the day. Tell them not to write on the cards!

2. Copy work sheets back-to-back for each student so that each page has six graphs. Working in pairs, each student will need five graphs, but this allows an extra in case they make a mistake. Each student needs his/her own work sheet. (If students are working individually, each student will need two back-to-back worksheets.)

3. Students solve each problem on their work sheet and record their answers.

4. Students cut the problems out and arrange the top row according to the letters on the template. (I cut out the class sets ahead of time.) Then, they must arrange the remaining boxes so that the solution is the midpoint of the two coordinate pair solutions directly above it. Paste all pieces down. (I have students show me their puzzle once they finish and I collect their work sheets.)

5. An answer key is provided. It is very easy to grade if you have the students cut and paste to complete the puzzle.


Partitioning a Line Segment Pyramid Puzzle

Directions: For each problem, find the coordinates of the point described. Record your answers in the ovals. Cut out the boxes and paste

E, G, B, and D on the top row (in that order) on the template. Paste the remaining boxes so that the solution to each problem is the midpoint

of the two ordered pair solutions directly above it.

A

Find the coordinates of P so

that P partitions 𝐴𝐵̅̅ ̅̅ in the ratio

2 to 3 with A (7, 2) and

B (-8, 9.5).

B

Given the points A (-1, 8) and

B (7, 10), find the coordinates of

the point P on the directed line

segment 𝐴𝐵̅̅ ̅̅ that partitions 𝐴𝐵̅̅ ̅̅

in the ratio 1:1.

C

Point A has coordinates (-9, -7).

Point B has coordinates (3, 8).

Find the coordinates of point P

that partition 𝐴𝐵̅̅ ̅̅ in the ratio

2:1.

D

For the directed line segment

whose endpoints are (4, 3) and

(-8, -5), find the coordinates of

the point that partitions the

segment into a ratio

of 3 to 1.

E

Given A (6, -10) and B (9, -1),

find the coordinates of point P

on 𝐴𝐵̅̅ ̅̅ so that P partitions 𝐴𝐵̅̅ ̅̅ in

the ratio 1:2 (i.e., so that AP:PB

is 1:2).

F



1 to 3 with A (2, -6) and

B (6, 6).

G

Given the points A (1, -0.5) and

B (-9, 7), find the coordinates of



in the ratio 1:4.

H





3:2.

I


whose endpoints are (7, 0.5) and

(-0.5, 3), find the coordinates of



of 4 to 1.

J

Given A (-10, -3.5) and B (2, 5.5),




is 5:1).


Partitioning a Line Segment Pyramid Puzzle Name ____________________ Period ___

E G B D


Partitioning a Line Segment Pyramid Puzzle Name ____________________ Period ___


Partitioning a Line Segment Pyramid Puzzle Solutions

E

Given A (6, -10) and B (9, -1),




is 1:2).

G

Given the points A (1, -0.5) and

B (-9, 7), find the coordinates of



in the ratio 1:4.

B

Given the points A (-1, 8) and

B (7, 10), find the coordinates of



in the ratio 1:1.

D


whose endpoints are (4, 3) and

(-8, -5), find the coordinates of



of 3 to 1.

F



1 to 3 with A (2, -6) and

B (6, 6).

A



2 to 3 with A (7, 2) and

B (-8, 9.5).

C





2:1.

H





3:2.

J

Given A (-10, -3.5) and B (2, 5.5),




is 5:1).

I


whose endpoints are (7, 0.5) and

(-0.5, 3), find the coordinates of



of 4 to 1.

(7, -7)

(1, 2.5)

(3, -3)

(2, 1) (0, 4)

(1, 5) (-1, 3)

(-5, -3) (3, 9) (-1, 1)







Documents

RATIOS - MR. ELKINS' GEOMETRY · Ratios can be shown in different ways: Use a “:” to separate the values: _____ Use the word “to” to separate the values: _____ PARTITIONING

RATIOS - MR. ELKINS' GEOMETRY · Ratios can be shown in different ways: Use a “:” to separate the values: _ Use the word “to” to separate the values: _ PARTITIONING