Download Lesson 2: Ratios and Coordinate Geometry Past knowledge:

Lesson 2
COMMON CORE MATHEMATICS CURRICULUM
U4
GEOMETRY
Name__________________________________________
Period______
Date____________
Lesson 2: Ratios and Coordinate Geometry
Past knowledge:
Use the picture (not drawn to scale) and determine the length of AC, CB, and AB. Assume all ratios or
fractions are 1 unit.
A
C
B
1. Line segment ̅̅̅̅
𝐴𝐵 is partitioned in the ratio 1:2. AC=_____, CB=_____, AB=______
2. Line segment ̅̅̅̅
𝐴𝐵 is partitioned in the ratio 7:5. AC=_____, CB=_____, AB=______
3. Line segment ̅̅̅̅
𝐴𝐶 is 2/3 of the distance of ̅̅̅̅
𝐴𝐵.
AC=_____, CB=_____, AB=______
Now change the assumption that we are working with 1 unit. Determine a method for solving the problems
below:
4. Line segment ̅̅̅̅
𝐴𝐵 is partitioned in the ratio 1:2. AC=_____, CB=_____, AB=18
̅̅̅̅ is partitioned in the ratio 7:5. AC=_____, CB=_____, AB=96
5. Line segment 𝐴𝐵
6. Line segment ̅̅̅̅
𝐴𝐶 is 2/3 of the distance of ̅̅̅̅
𝐴𝐵.
AC=_____, CB=_____, AB=96
7. A board 24 inches long is cut into two pieces in the ratio 1:2. Find the length of each piece.
8. Discuss how you approached the ratio problems.
9. Discuss how you approached the fraction problems.
1
COMMON CORE MATHEMATICS CURRICULUM
Lesson 2
U4
GEOMETRY
Part 1: Partitioning a Segment – How to find the point on a segment that
partitions the segment by a given ratio based on coordinates?
10. Given the points A(-5, 0) and B(5, 0), find the coordinates of the point P on directed line segment ̅̅̅̅
𝐴𝐵 that
partitions ̅̅̅̅
𝐴𝐵 in the ratio 1:1.
How did you determine the coordinates of this midpoint?_____________________________
̅̅̅̅
11. Given the points A(-5, 0) and B(10, 0), find the coordinates of the point P on directed line segment 𝐴𝐵
̅̅̅̅
that partitions 𝐴𝐵 in the ratio 2:3.
̅̅̅̅ into two
12. Given the points A(-5, 0) and B(10, 0), what is the other location for point P so that P divides 𝐴𝐵
parts with lengths in a ratio of 2:3. Note, since this is not directed, you can do it from either side.
13. Given the points A(-3, -5) and B(7, 0), find the two possible coordinates of the point P on line segment ̅̅̅̅
𝐴𝐵
̅̅̅̅ in the ratio 2:3.
that partitions 𝐴𝐵
2
COMMON CORE MATHEMATICS CURRICULUM
Lesson 2
U4
GEOMETRY
̅̅̅̅
14. Given the points A(–1, 2) and B(7, 14), find two possible coordinates of the point P on line segment 𝐴𝐵
̅̅̅̅
that partitions 𝐴𝐵 in the ratio 1:3.
15. Given the points A(-2, 4) and B(7, -2), find two possible coordinates of the point P on line segment ̅̅̅̅
𝐴𝐵
̅̅̅̅ in the ratio 1:2.
that partitions 𝐴𝐵
16. Line segment ̅̅̅
𝐽𝐾 in the 𝑥𝑦-coordinate plane has endpoints with coordinates (−4, 11) and (8, −1). What
are two possible locations for point M so that M divides ̅̅̅
𝐽𝐾 into two
parts with lengths in a ratio of 1: 3?
3
COMMON CORE MATHEMATICS CURRICULUM
Lesson 2
U4
GEOMETRY
17. While visiting a daycare center, you estimate the ratio of toddlers to infants as 3:2. If the center has an
enrollment of 30 children, about how many of them are infants?
Use the map and the information given to solve each problem that follows.
18. Luis works at a theater on 8th Avenue and 20th Street. Kaleb lives at the corner of 18th Avenue and 4th
Street. What is a possible location that is midway between them?
19. Nima lives at the corner of 4th Avenue and 4th Street. Bill lives at the corner of 10th Avenue and 6th
Street. Their favorite bakery is located midway between them. What is one possible location of the
bakery?
4
COMMON CORE MATHEMATICS CURRICULUM
Lesson 2
U4
GEOMETRY
20. Cleve’s Cookie Store is located at the corner of 2nd Avenue and 9th Street. Dave’s Doorknobs is located
at the corner of 12th Avenue and 14th Street. Located 1/5 of the distance from Cleve’s Cookie Store is
the post office. Where is the post office?
21. Malik and Brad both live on 3rd Avenue. Malik lives at the corner of 1st Street, and Brad lives at the
corner of 19th Street. 2/3 the distance from Malik’s apartment to Brad’s apartment is a market. Where
is the market?
22. The main entrance to the high school is located at the corner of 17th Avenue and 19th Street. On his way
from school to the bank, Luis stops at the coffee shop located at 12th Avenue and 15th Street. The coffee
shop is the midpoint of this trip. What is the location of the bank?
Angles and Shapes:
23. The perimeter of rectangle ABCD is 60 cm. The ratio of AB:BC is 3:2. Find the length and width of the
rectangle.
w
B
C
l
A
D
24. The perimeter of the isosceles triangle shown is 56 in. The ratio of LM:MN is 5:4. Find the lengths of the
sides and the base of the triangle.
L
N
M
5
COMMON CORE MATHEMATICS CURRICULUM
Lesson 2
U4
GEOMETRY
Part 2: Extended Ratios (Comparing more than two items)
25. The measure of the angles in ∆JKL are in the extended ratio of 1:2:3. Find the measures of the angles.
(Draw a sketch)
26. The measures of the angles in a triangle are in the extended ratio 3:4:8. Find the measures of the angles.
(draw a sketch)
More practice. These all must be complete:
27. The measure of the angles in ∆JKL are in the extended ratio of 1:7:10. Find the measures of the angles.
(Draw a sketch)
28. The measure of the angles in ∆JKL are in the extended ratio of 1:9:10. Find the measures of the angles.
(Draw a sketch)
29. The measure of the angles in ∆JKL are in the extended ratio of 7:16:22. Find the measures of the angles.
(Draw a sketch)
Find the point that partitions the segment with the two given endpoints with the given
ratio. Provide all answers. You can share a graph between 2 problems if
you want to use it.
30. (-3, 4) (7, 6) 1:1
31. (-9, 3) (1, 8) 2:3
6
COMMON CORE MATHEMATICS CURRICULUM
Lesson 2
U4
GEOMETRY
32. (8, -5) (4, 7) 1:3
33. (5, -6) (0, 9) 3:2
34. A triangle with a perimeter of 63 feet has side lengths in the extended ratio of 6 : 7 : 8. Find the side
lengths of the triangle.
35. A board 18 inches long is cut into two pieces in the ratio 2:7. Find the length of each piece.
7