CC-1 Partitioning a Common Core State Standards Find the...
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You can find a point on a line segment such that the point’s distance from one endpoint of the segment and the length of the entire segment have a specified ratio. On a number line, A is at −2 and B is at 4. What is the location of C between A and B, such that AC is 2 3 the length of AB ? 1. Find the length of AB . 2. What is 2 3 the length of AB ? 3. What is the location of point C that is 2 3 the length of AB from A to B? 4. Find the ratio AC AB . Explain how this checks your solution. 5. a. On the vertical number line at the right, find point E such that DE = 3FD. Explain your work. b. Is there another value of E such that DE = 3FD? Explain. Points A( − 2, − 3) and B(8, 2) are the endpoints of AB . What are the coordinates of point C on AB such that AC is 2 5 the length of AB ? 6. Find 2 5 of the difference between the x-coordinates of A and B. 7. Find 2 5 of the difference between the y-coordinates of A and B. 8. How many units to the right or left of point A is point C located? How many units up or down from point A is point C located? 9. Find the coordinates of point C. 1 B A 0 1 2 3 4 5 1 2 3 4 5 D F 0 1 2 3 4 5 6 1 2 3 4 5 6 2 x y 2 4 6 8 10 2 4 6 8 10 4 6 8 10 O 4 6 8 10 B A Partitioning a Line Segment CC - 1 Common Core State Standards MACC.912.G-GPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. MP 5 CC-1 Partitioning a Line Segment 1 CC-1 1 Answers Activity 1 1. 6 units 2. 4 units 3. 2 4. AC AB = 4 6 = 2 3 ; AC is 2 3 the length of AB . 5a. - 2 FD = 2, so 3FD = 3(2) = 6. Point D is located at 4, so point E is located at 4 - 6, or –2. b. Point E could also be located at 4 + 6 = 10 because DE = 6, which equals 3FD. Activity 2 6. 2 5 (8 - (–2)) = 2 5 (10) = 4 units 7. 2 5 (2 - (–3)) = 2 5 (5) = 2 units 8. 4 units to the right of A; 2 units up from A 9. (2, –1) 10. AC = 2 25 ; AB = 5 25 ; AC AB = 2 15 515 = 2 5 ; AC is 2 5 the length of AB from A. 11. Yes; if point C is 2 5 the length of AB from A, then AC = 2 5 AB and CB = 3 5 AB because AC + CB = AB. 12. The midpoint M between A(x 1 , y 1 ) and B(x 2 , y 2 ) is M ( 1 2 (x 1 + x 2 ), 1 2 (y 1 + y 2 ) ) . To find a point P, p q of the length of AB from an endpoint, you find p q of the difference in x-coordinates and p q of the difference in y-coordinates and add or subtract the results from the coordinates of the endpoint. Guided Instruction PURPOSE To partition a line segment into a given ratio PROCESS Students will • find a point on a number line that divides a line segment into a given ratio. • find a point on a line segment in the coordinate plane that divides the line segment into a given ratio. DISCUSS Have students describe how to find 2 3 of a whole. Have them describe how they would find 2 3 of any number other than 1. Remind students that when working with a number line or in the coordinate plane, they need to know the length of the line segment to determine the given ratio. Activity 1 ERROR PREVENTION Show students how 2 3 of AB from A is not the same as 2 3 of AB from B. From A, point C will be located at 2. From B, point C will be located at 0. Activity 2 Mathematical Practices CC-1 supports students in becoming proficient in using available tools, including diagrams, to solve a mathematical problem, Mathematical Practice 5. Q What do you need to determine first? [The distance between points A and B, or the length of the line segment.] Q After you determine 2 3 of AB, how do you locate point C? [Add 4 to –2.] Q Why don’t you need to use the distance formula to find the coordinates of point C? [You can multiply 2 5 by the difference in the x-coordinates and add that number to the x-coordinate of point A to find the x-coordinate of point C. Then you can repeat the process for the y-coordinates.]
Transcript of CC-1 Partitioning a Common Core State Standards Find the...
Concept Byte X-X Repl title on H-CB_base_open 1
You can find a point on a line segment such that the point’s distance from one endpoint of the segment and the length of the entire segment have a specified ratio.
On a number line, A is at −2 and B is at 4. What is the location of C between A and B, such that AC is 23 the length of AB?
1. Find the length of AB.
2. What is 23 the length of AB?
3. What is the location of point C that is 23 the length of AB from A to B?
4. Find the ratio ACAB . Explain how this checks your solution.
5. a. On the vertical number line at the right, find point E such that DE = 3FD. Explain your work.
b. Is there another value of E such that DE = 3FD? Explain.
Points A(−2, −3) and B(8, 2) are the endpoints of AB. What are the coordinates of point C on AB such that AC is 25 the length of AB?
6. Find 25 of the difference between the x-coordinates of A and B.
7. Find 25 of the difference between the y-coordinates of A and B.
8. How many units to the right or left of point A is point C located? How many units up or down from point A is point C located?
9. Find the coordinates of point C.
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Partitioning a Line Segment
CC-1 Common Core State StandardsMACC.912.G-GPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
MP 5
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CC-1 1
AnswersActivity 1 1. 6 units 2. 4 units 3. 2 4. AC
AB = 46 = 2
3; AC is 23 the length of AB.
5a. - 2 FD = 2, so 3FD = 3(2) = 6. Point D is located at 4, so point E is located at 4 - 6, or –2.
b. Point E could also be located at 4 + 6 = 10 because DE = 6, which equals 3FD.
Activity 2 6. 2
5(8 - (–2)) = 25(10) = 4 units
7. 25(2 - (–3)) = 2
5(5) = 2 units 8. 4 units to the right of A; 2 units up
from A 9. (2, –1)
10. AC = 225; AB = 525; ACAB =
215515
= 25; AC is 25 the length of
AB from A. 11. Yes; if point C is 25 the length of AB
from A, then AC = 25AB and CB =
35 AB because AC + CB = AB.
12. The midpoint M between A(x1, y1) and B(x2, y2) is
M(12(x1 + x2), 12(y1 + y2)) . To find
a point P, pq of the length of AB
from an endpoint, you find pq of the
difference in x-coordinates and pq of
the difference in y-coordinates and add or subtract the results from the coordinates of the endpoint.
Guided InstructionPURPOSE To partition a line segment into a given ratioPROCESS Students will•findapointonanumberlinethatdividesaline
segment into a given ratio.•findapointonalinesegmentinthecoordinate
plane that divides the line segment into a given ratio.
DISCUSS Have students describe how to find 23 of a whole. Have them describe how they would find 23 of any number other than 1.Remind students that when working with a number line or in the coordinate plane, they need to know the length of the line segment to determine the given ratio.
Activity 1
ERROR PREVENTIONShow students how 23 of AB from A is not the same as 23 of AB from B. From A, point C will be located at 2. From B, point C will be located at 0.
Activity 2
Mathematical Practices CC-1 supports students in becoming proficient in using available tools, including diagrams, to solve a mathematical problem, Mathematical Practice 5.
Q What do you need to determine first? [The distance between points A and B, or the length of the line segment.]
Q After you determine 23 of AB, how do you locate
point C? [Add 4 to –2.]
Q Why don’t you need to use the distance formula to find the coordinates of point C? [You can multiply 25 by the difference in the x-coordinates and add that number to the x-coordinate of point A to find the x-coordinate of point C. Then you can repeat the process for the y-coordinates.]
2 Concept Byte Justifying Constructions
10. To check your solution, use the distance formula to find AC and AB. Find ACAB .
Explain how this checks your solution.
11. Is finding a point 25 the length of AB from A the same as finding a point 35 the length of AB from B? Explain.
12. How does this process compare to using the midpoint formula?
Exercises 13. On a number line, A is at -3 and B is at 5. What is the location of C such that AC is
23 the length of AB from A to B?
14. Points A(-1, 2) and B(5, 16) are the endpoints of AB. What are the coordinates of point C on AB such that AC is 23 the length of AB from B?
15. Use the number line shown below. What is the location of Z between X and Y such that the length of ZY is 3 times the length of XZ?
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16. Points X(6, 4) and Y(-4, -16) are the endpoints of XY . What are the coordinates of point Z on XY such that XZ is 45 the length of XY ?
17. The graph below shows UV . What are the coordinates of point W on UV such that W is 56 the length of UV from V?
18. A has coordinates (x1, y1) and B has coordinates (x2, y2).
a. Write a formula to find point C on AB such that AC is pq the length of AB.
Explain why your formula works. b. What is a formula to find point C on AB such that the ratio of AC to BC is p to q?
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Concept Byte Parallel Lines and Related Angles 3
Complementary Angles and Trigonometric Ratios
The acute angles of a right triangle are complementary because the sum of their measures is 90°. There is a relationship between the sine and cosine of complementary angles.
What is the relationship between the sine and cosine of complementary angles?
1. Refer to right triangle PQR, shown at the right.
a. How is the opposite side of ∠R related to the adjacent side of ∠Q? b. How is the adjacent side of ∠R related to the opposite side of ∠Q? c. Compare sin R and cos Q. d. Compare sin Q and cos R. e. Make a conjecture based on your results from parts (c) and (d).
2. Refer to right triangle ABC, shown at the right.
a. Are angles B and C complementary? Justify your answer. b. Write sine and cosine ratios for angles B and C. What do you notice about the
relationship between sin B and cos C? Between sin C and cos B? c. Is the statement sin B = cos C true for all complementary angles of a right
triangle? Explain.
3. Reasoning Prove that in a right triangle with acute angles B and C, cos B = sin (90° - m∠B).
Exercises 4. In the figure at the right, line c is perpendicular to line d. a. For △LMN, cos L is equal to the sine of which angle? Explain. b. For △MOP, sin P is equal to the cosine of which angle? Explain. c. For △MRQ, cos Q is equal to the sine of which angle? Explain.
5. The sine of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. If you examine the structure of the word cosine and the relationship of the angles in a right triangle, how do you think cosine might have gotten its name?
6. A taut wire runs from the top of a tall pole to the ground. The pole is perpendicular to the ground. The sine of the angle that the wire makes with the ground is 12
13. What is the cosine of the angle that the wire makes with the top of the pole? Explain.
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CC-2 Common Core State StandardsMACC.912.G-SRT.3.7 Explain and use the relationship between the sine and cosine of complementary angles.
MP 3
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2 Common Core
Exercises 13. 21
3 14. (1, 62
3) 15. –23
4 16. (–2, –12) 17. (–3, 23)
18a. (x1 +pq (x2 - x1), y1 +
pq (y2 - y1)).
The formula works because you are finding
pq of the x-distance and
pq of
the y-distance from the endpoint A and adding the results to the coordinates of A.
b. (x1 +p
p + q (x2 - x1),
y1 + pp + q (y2 - y1))
