HSG.GPE.B.6: Partitioning a Directed Line Segment in a Given Ratio
In plain English: HSG.GPE.B.6 is the Common Core geometry standard that asks students to find the point on a directed line segment between two given points that partitions the segment in a given ratio. Students turn a ratio such as 2:3 into the fraction 2/5 of the way from the starting point and apply it to the changes in x and y. It is usually taught in Geometry.
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Common Core State Standards for Mathematics · Domain: Expressing Geometric Properties with Equations (GPE) · Cluster: Use coordinates to prove simple geometric theorems algebraically Also written as HSG-GPE.B.6 or G-GPE.6 · Official standard
Students learn to find the point on a directed line segment that partitions it in a given ratio. The lesson starts on a number line, where the difference between a ratio of parts (such as 1:3) and a fraction of the way (such as 1/3) is easiest to see, and then moves to the coordinate plane. Students turn the ratio a:b into the fraction a/(a + b), apply it to the changes in x and y from the starting point, and check the result.
Similar slope triangles explain why the same fraction applies to both coordinates. Students also see that direction matters: partitioning from A to B and from B to A in the same ratio gives different points. The lesson ends with problems set on maps and routes, where the partition point has a real meaning and the distances can be checked.
Learning Objectives
By the end of this lesson, students will be able to:
Convert a ratio a:b into the fraction a/(a + b) of the way from the starting point of a directed segment
Find the point that partitions a directed segment on a number line or in the coordinate plane in a given ratio
Explain how the direction of the segment changes the partition point
Explain with similar triangles why the same fraction applies to the x- and y-coordinates
Check a partition point with the distance formula and interpret it in context
Prior Knowledge Required
Students should already be comfortable with:
Ratios and proportional relationships 7.RP.A.2
Finding the distance between two points 8.G.B.8
Slope triangles and similar triangles on the coordinate plane 8.EE.B.6
Draw a number line from 0 to 16 and mark A at 3 and B at 15.
Warm-Up Prompt
"Where is the point that is 1/3 of the way from A to B? Where is the point that splits AB into two pieces whose lengths are in the ratio 1:3? Are these the same point? Why or why not?"
AB has length 12. One third of the way is 3 + 4 = 7. The ratio 1:3 means 1 part to 3 parts, 4 parts in all, so the point is 1/4 of the way: 3 + 3 = 6. Many students give 7 for both questions. Use the disagreement to introduce the two key ideas of the lesson: a ratio a:b compares the two pieces, so the point is a/(a + b) of the way, and the segment is directed, so "from A to B" tells you where to start counting.
Direct Instruction20 minutes
The method. A directed segment from A(x₁, y₁) to B(x₂, y₂) starts at A and ends at B. The point P that partitions it in the ratio a:b satisfies PA : PB = a : b. Teach these steps:
Name the start and the end. "From A to B" means start at A. The ratio is measured from the start.
Turn the ratio into a fraction of the way. The segment is cut into a + b equal parts, and P is a of them from the start, so k = a/(a + b).
Find the change from start to end. Δx = x₂ - x₁ and Δy = y₂ - y₁. Keep the signs.
Move k of the way. P = (x₁ + kΔx, y₁ + kΔy).
Check. P should be between A and B, closer to the start when a < b, and the lengths PA and PB should be in the ratio a:b.
Explain why it works with Diagram 1: the slope triangle from A to B is split by P into two similar triangles. Their horizontal sides are in the ratio 3:2 and so are their vertical sides, so the same fraction applies to Δx and to Δy. The same idea gives an equivalent weighted-average form, P = ((b·x₁ + a·x₂)/(a + b), (b·y₁ + a·y₂)/(a + b)), where the endpoint farther from P gets the smaller weight. The midpoint formula is the case a = b.
On a number line
Find the point that partitions the directed segment from A = -4 to B = 11 in the ratio 2:3.
Equation: k = 2/5, length 15, so P = -4 + (2/5)(15) = 2.
In the coordinate plane
Find P on the directed segment from A(-2, -1) to B(8, 4) that partitions it in the ratio 3:2.
Equation: k = 3/5: P = (-2 + (3/5)(10), -1 + (3/5)(5)) = (4, 2).
Direction matters
Same segment, but from B(8, 4) to A(-2, -1) in the ratio 3:2.
Equation: Start at B: P = (8 + (3/5)(-10), 4 + (3/5)(-5)) = (2, 1).
Fraction of the way
Find the point 1/4 of the way from C(1, 9) to D(9, -3). What ratio is this?
Equation: P = (1 + (1/4)(8), 9 + (1/4)(-12)) = (3, 6); the ratio is 1:3.
In context
On a town map in kilometers, a straight bike trail runs from (2, 1) to (14, 10). A rest stop divides the trail from the start in the ratio 2:3. Where is it?
Equation: k = 2/5: (2 + 4.8, 1 + 3.6) = (6.8, 4.6). The trail is 15 km long, so the stop is 6 km from the start.
After Example 3, compare P(4, 2) and P(2, 1) on Diagram 1: the same segment and the same ratio give different points when the direction changes. Diagram 2 shows the same idea for the ratio 1:3.
Guided Practice15 minutes
Pairs solve four problems. For each, one partner says the start point and the fraction k before either partner computes.
A(1, -2) to B(13, 6) in the ratio 1:3. (k = 1/4: (4, 0).)
The same segment from B to A in the ratio 1:3. (Start at B: (10, 4).)
E(-6, 2) to F(4, -3) in the ratio 4:1. (k = 4/5: (2, -2).)
G(3, -5) to H(3, 7) in the ratio 5:1. (The segment is vertical: (3, 5).)
Listen for these errors: using a/b instead of a/(a + b), starting from the wrong endpoint, and dropping the negative sign in Δx or Δy.
Independent Practice10-15 minutes
Students work alone on five problems and check each answer with the distance formula or a sketch:
J(-5, 4) to K(7, -2) in the ratio 2:1. ((3, 0).)
M(0, 8) to N(10, 3) in the ratio 3:2. ((6, 5).)
N(10, 3) to M(0, 8) in the ratio 3:2. ((4, 6).)
On a number line, from -9 to 6 in the ratio 4:1. (3.)
The point 2/3 of the way from R(4, -1) to S(-5, 5). ((-2, 3).)
Closure5 minutes
Exit ticket: For A(2, 3) and B(10, -1), (1) find the point that partitions the directed segment from A to B in the ratio 1:1 ((6, 1), the midpoint), (2) find the point for the ratio 3:1 ((8, 0)), and (3) explain in one sentence why the ratio 3:1 does not mean 1/3 of the way.
Differentiation Strategies
For Struggling Students
Start on a number line and draw the a + b equal parts before computing anything
Use a table with columns "start," "change," "k times change" and "start + k times change" for x and for y
Begin with horizontal and vertical segments, then move to slanted segments with changes that divide evenly
For Advanced Students
Derive the weighted-average form P = (bA + aB)/(a + b) from P = A + (a/(a + b))(B - A)
Given P(1, 2) partitions the directed segment from A(-3, 0) to B in the ratio 2:3, find B, and explain each step
Show that the point partitioning AB in the ratio a:b is the image of B under the dilation centered at A with scale factor a/(a + b)
Assessment Guidance
What to Look For
Check that students write the fraction k = a/(a + b) before computing, and that they start from the first-named endpoint. A partition point that is not between the endpoints, or that is closer to the wrong endpoint, points to a direction or fraction error. In context problems, look for an answer that uses the units and a quick distance check, such as "the stop is 6 km from the start and 9 km from the end, and 6:9 = 2:3."
02
Classroom Activities
3 Activities
1
Human Number Line
15 minWhole class, then pairs
A 12-foot strip of masking tape on the floor becomes a directed segment from A (0 ft) to B (12 ft). Students stand where they think each partition point is, then measure to check. Moving their bodies makes the difference between ratios and fractions, and the role of direction, concrete.
Prompts (with answers for the teacher)
From A to B in the ratio 1:2 (4 ft)
From A to B in the ratio 2:1 (8 ft)
From A to B in the ratio 5:1 (10 ft)
From A to B in the ratio 1:1 (6 ft, the midpoint)
From A to B in the ratio 3:1 (9 ft)
From B to A in the ratio 1:2 (8 ft from A, because the count starts at B)
Procedure
For each prompt, three volunteers stand at their guesses, and the class votes before a student measures with the tape measure
After the whole-class round, pairs draw the same segment on graph paper and repeat the prompts with A at 0 and B at 12
Pairs write a rule for the position of P in terms of a, b and the length 12
Modification for Distance Learning
Share a number line from 0 to 12 on a slide, and have students drag a marker to each partition point. Reveal the answers after each prompt.
2
Partition Card Match
20 minPairs
Each pair receives 8 problem cards and 11 answer cards: the 8 correct points and 3 decoys that come from common errors. Pairs match each problem to its point and set aside the decoys, naming the error behind each one.
Problem Cards (with answers for the teacher)
Card 1: A(0, 0) to B(9, 6), ratio 1:2 ((3, 2))
Card 2: A(0, 0) to B(9, 6), ratio 2:1 ((6, 4))
Card 3: B(9, 6) to A(0, 0), ratio 1:2 ((6, 4), the same point as Card 2)
Card 4: C(-4, 3) to D(6, -2), ratio 3:2 ((2, 0))
Card 5: C(-4, 3) to D(6, -2), ratio 1:4 ((-2, 2))
Card 6: E(1, -7) to F(1, 5), ratio 1:3 ((1, -4))
Card 7: G(-8, -2) to H(4, -2), ratio 5:1 ((2, -2))
Card 8: J(2, 10) to K(-6, -2), ratio 3:1 ((-4, 1))
Decoy Answer Cards
(0, 1): Card 4 computed from D to C instead of from C to D
(1, 2): Card 6 computed from F to E
(4.5, 3): the midpoint of Card 1, from reading 1:2 as one half
Discussion Questions
Why do Cards 2 and 3 give the same point?
What single change to Card 4 would produce the decoy (0, 1)?
How can you tell, without computing, that (4.5, 3) cannot be the answer to Card 1?
3
Trail Map Planning
15-20 minGroups of 3-4
Groups plan facilities along two straight trails on a park map where 1 grid unit is 1 km. For each facility, they find its coordinates and its distance along the trail, and then check that the distances are in the required ratio.
The Tasks (with answers for the teacher)
Trail 1 runs from the trailhead T(0, 0) to the summit U(16, 12). Water stations divide the trail from T in the ratios 1:3 and 3:1. ((4, 3) and (12, 9); the trail is 20 km long, so they are 5 km and 15 km from T.)
A ranger station divides Trail 1 from U to T in the ratio 2:3. ((9.6, 7.2), 8 km from U.)
Trail 2 runs from V(4, 14) to W(19, 6). A lookout divides it from V in the ratio 2:3. ((10, 10.8); the trail is 17 km long, so the lookout is 6.8 km from V.)
Procedure
Groups plot both trails on graph paper and mark each facility
For every facility, the group checks its answer with the distance formula and states the two distances along the trail
Groups present one facility and explain how they chose the starting point
Challenge Variation
The park wants 4 benches that split Trail 2 into 5 equal pieces. Groups find all four points and describe the pattern in their coordinates.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Partitioning a Directed Segment in the Ratio 3:2
The directed segment from A(-2, -1) to B(8, 4) is cut into 5 equal parts. P(4, 2) is 3 parts from A and 2 parts from B, so PA : PB = 3 : 2. The dashed slope triangle shows why the same fraction 3/5 applies to both coordinates: the run of 10 splits into 6 and 4, and the rise of 5 splits into 3 and 2. Drawn to scale.
Diagram 2: The Direction of the Segment Changes the Point
The same segment and the same ratio 1:3 give two different points. From A to B, P(-1, 3) is 1/4 of the way from A. From B to A, Q(3, -1) is 1/4 of the way from B. Drawn to scale.
Directions: For each problem, name the starting point, write the fraction k = a/(a + b), and show the change in each coordinate. Check at least three of your answers with the distance formula or a sketch on graph paper.
Part 1: Number Lines and Vertical Segments (Problems 1-2)
On a number line, A = -7 and B = 13. Find the point that partitions the directed segment from A to B in the ratio 3:2. Then find the point that partitions the directed segment from B to A in the ratio 3:2. Explain why the answers are different.
Find the point that partitions the directed segment from P(-4, -6) to Q(-4, 9) in the ratio 2:3.
Part 2: Segments in the Coordinate Plane (Problems 3-5)
Find the point that partitions the directed segment from A(-3, 7) to B(9, -1) in the ratio 1:3.
Find the point that partitions the directed segment from C(2, -5) to D(-8, 10) in the ratio 4:1. Is your point closer to C or to D? Explain how you could have predicted this.
For E(5, 2) and F(-1, -7), find (a) the point 2/3 of the way from E to F and (b) the point that partitions the directed segment from E to F in the ratio 2:3. Explain why these are different points.
Part 3: Application (Problem 6)
On a map where 1 unit is 1 km, a straight water pipeline runs from a pump station at W(3, 2) to a reservoir at R(27, 20). An inspection valve divides the pipeline from W to R in the ratio 5:1. Find the coordinates of the valve and its distance from each end of the pipeline.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Fraction k
k = a/(a + b) written correctly for every problem
One ratio treated as a/b or as a fraction of the whole
Fractions missing or incorrect
Direction
Always starts from the first-named endpoint
One direction error
Direction ignored
Computation
All coordinates correct, with signs
One or two arithmetic errors
Many errors
Explanation and Context
Explanations clear; Problem 6 gives both distances with units
Explanations incomplete or units missing
No explanations
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
On a number line, A = 2 and B = 20. Which point partitions the directed segment from A to B in the ratio 1:2?
Answer: A
The segment has length 18, and the ratio 1:2 means P is 1/3 of the way from A: 2 + (1/3)(18) = 8. Check: PA = 6 and PB = 12, and 6:12 = 1:2. Choice B uses the ratio 2:1. Choice C is the midpoint. Choice D is (1/3)(18) without adding the starting value 2.
Question 2 of 20 · Multiple Choice
Point P partitions the directed segment from A to B in the ratio 3:5. What fraction of the way from A to B is P?
Answer: C
The ratio 3:5 splits AB into 3 + 5 = 8 equal parts, and P is 3 of them from A, so it is 3/8 of the way. Choice A treats the ratio of the parts as a fraction of the whole, a common error. Choice B measures from B instead of A.
Question 3 of 20 · Multiple Choice
Which point partitions the directed segment from A(0, 0) to B(15, 5) in the ratio 2:3?
Answer: B
k = 2/(2 + 3) = 2/5, so P = (0 + (2/5)(15), 0 + (2/5)(5)) = (6, 2). Choice A uses k = 3/5, which measures the ratio from B. Choice C is the midpoint. Choice D uses k = 2/3, the ratio of the two parts, as if it were a fraction of the whole.
Question 4 of 20 · Multiple Choice
Which point partitions the directed segment from A(-4, 1) to B(8, 7) in the ratio 1:2?
Answer: C
k = 1/3, Δx = 12 and Δy = 6, so P = (-4 + 4, 1 + 2) = (0, 3). Choice A uses k = 2/3 (the ratio 2:1). Choice B is the midpoint. Choice D uses k = 1/4, as if the ratio were 1:3.
Question 5 of 20 · Multiple Choice
A(1, -3) and B(9, 5) are given. Which point partitions the directed segment from B to A in the ratio 1:3?
Answer: D
The segment starts at B. k = 1/4, and the change from B to A is (-8, -8), so P = (9 - 2, 5 - 2) = (7, 3). Choice A starts at A instead of B. Choice B is the midpoint. Choice C is outside the segment, from moving 1/4 of the change past A.
Question 6 of 20 · Multiple Choice
Which expression gives the point that partitions the directed segment from A(x₁, y₁) to B(x₂, y₂) in the ratio a:b?
Answer: B
The point is a/(a + b) of the way from the start A, so each coordinate is the starting value plus that fraction of the change. Choice A uses a/b, which can even land past B when a > b. Choice C starts at B, so it partitions the segment from B to A. Choice D is the midpoint, which is only the case a = b.
Question 7 of 20 · Multiple Choice
Point P partitions the directed segment from A to B in the ratio 2:1. Which statement is true?
Answer: A
The ratio 2:1 means PA : PB = 2 : 1, so PA = 2·PB and P is 2/3 of the way from A. Choice B reverses the ratio. Choice D would put P at the midpoint, which is the ratio 1:1.
Question 8 of 20 · Multiple Choice
Which point partitions the directed segment from (5, -8) to (5, 4) in the ratio 3:1?
Answer: A
The segment is vertical, so only y changes. k = 3/4 and Δy = 12, so y = -8 + 9 = 1. Choice B uses k = 1/4 (the ratio 1:3). Choice C is the midpoint. Choice D uses k = 1/3.
Question 9 of 20 · Multiple Choice
Which point partitions the directed segment from A(-6, -3) to B(3, 9) in the ratio 2:1?
Answer: D
k = 2/3, Δx = 9 and Δy = 12, so P = (-6 + 6, -3 + 8) = (0, 5). Choice A uses k = 1/3. Choice B is the midpoint. Choice C is the change k(Δx, Δy) = (6, 8) without adding the starting point.
Question 10 of 20 · Multiple Choice
Point P partitions the directed segment from A to B in the ratio 3:2, and AB = 20. What is PA?
Answer: B
P is 3/5 of the way from A, so PA = (3/5)(20) = 12 and PB = 8. Choice A is PB. Choice C treats the ratio as a third of the segment. Choice D uses 3/4.
Question 11 of 20 · Multiple Choice
A(2, 8) and B(17, 3) are given. The point P = (1·A + 4·B)/5 partitions the directed segment from A to B in the ratio 4:1. What is P?
Answer: D
P = ((1·2 + 4·17)/5, (1·8 + 4·3)/5) = (70/5, 20/5) = (14, 4). The check P = A + (4/5)(B - A) = (2 + 12, 8 - 4) gives the same point. Choice A swaps the weights, (4·A + B)/5, which gives the point for the ratio 1:4. Choice B is the midpoint, and choice C is the point for the ratio 3:2.
Question 12 of 20 · Multiple Choice
For A(0, 0) and B(12, 6), a student says the point that partitions the directed segment from A to B in the ratio 1:2 is (6, 3). What is the error?
Answer: A
(6, 3) is the midpoint, so the student moved 1/2 of the way. The ratio 1:2 has 3 parts in all, so k = 1/3 and P = (4, 2). Choice B would give (8, 4), which is 1/3 of the way from B.
Question 13 of 20 · Multiple Choice
On a map in kilometers, a hiker walks a straight route from (1, 2) to (13, 11). She stops at the point that divides the route from the start in the ratio 1:2. How far has she walked?
Answer: C
The route has length √(12² + 9²) = 15 km, and the stop is 1/3 of the way, so she has walked 5 km. The stop is at (5, 5). Choice A is half the route. Choice B is 2/3 of the route, the ratio 2:1. Choice D is 1/4 of the route.
Question 14 of 20 · Multiple Choice
Which point partitions the directed segment from C(-2, 6) to D(10, -6) in the ratio 5:1?
Answer: C
k = 5/6, Δx = 12 and Δy = -12, so P = (-2 + 10, 6 - 10) = (8, -4). Choice A starts at D instead of C. Choice B is the midpoint. Choice D uses a/b = 5/4, which puts the point beyond D, outside the segment.
Question 15 of 20 · Short Answer
Find the point that partitions the directed segment from A(-5, -2) to B(7, 7) in the ratio 2:1.
k = 2/3, Δx = 12 and Δy = 9. P = (-5 + (2/3)(12), -2 + (2/3)(9)) = (-5 + 8, -2 + 6) = (3, 4).
Question 16 of 20 · Short Answer
A(3, -4) and B(-7, 11) are given. Find the point that partitions the directed segment from B to A in the ratio 1:4.
Start at B. k = 1/5, and the change from B to A is (10, -15). P = (-7 + 2, 11 - 3) = (-5, 8). It is closer to B, as expected for the ratio 1:4 measured from B.
Question 17 of 20 · Short Answer
Find the point 3/4 of the way from M(-8, 5) to N(4, -3). In what ratio does this point partition the directed segment from M to N?
The change is (12, -8), so P = (-8 + 9, 5 - 6) = (1, -1). Three quarters of the way means 3 parts out of 4, so the ratio is 3:1.
Question 18 of 20 · Short Answer
On a number line, find the point that partitions the directed segment from 17 to -3 in the ratio 3:1.
The change from 17 to -3 is -20, and k = 3/4. P = 17 + (3/4)(-20) = 17 - 15 = 2. Check: the distance from 17 to 2 is 15 and from 2 to -3 is 5, and 15:5 = 3:1.
Question 19 of 20 · Short Answer
A straight fence runs from post F(0, 0) to post G(24, 7), with coordinates in meters. A gate is placed so that it divides the fence from F to G in the ratio 3:2. Find the location of the gate and its distance from F.
k = 3/5, so the gate is at ((3/5)(24), (3/5)(7)) = (14.4, 4.2). The fence is √(24² + 7²) = 25 m long, so the gate is 15 m from F and 10 m from G.
Question 20 of 20 · Short Answer
Show that P(4, 1) partitions the directed segment from A(-2, -3) to B(13, 7) in the ratio 2:3, first with the fraction method and then with the distance formula.
Fraction method: k = 2/5 and the change is (15, 10), so the point is (-2 + 6, -3 + 4) = (4, 1). Distance check: PA = √(6² + 4²) = √52 = 2√13 and PB = √(9² + 6²) = √117 = 3√13. PA : PB = 2√13 : 3√13 = 2 : 3, so P partitions the segment in the ratio 2:3.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSG.GPE.B.6 mean?
It means students must find the point between two given points that splits the segment joining them in a given ratio, measured from a given starting point. For example, the point that partitions the directed segment from A(1, 1) to B(13, 7) in the ratio 1:2 is (5, 3), because it is 1/3 of the way from A.
Is HSG.GPE.B.6 taught in Geometry or Algebra?
HSG.GPE.B.6 is a Geometry standard and is usually taught in Geometry, in the coordinate geometry unit together with the distance and midpoint formulas. It uses ratio reasoning from middle school and signed arithmetic from Algebra I.
What is the formula for partitioning a segment in a given ratio?
For the directed segment from A(x₁, y₁) to B(x₂, y₂) and the ratio a:b, let k = a/(a + b). Then P = (x₁ + k(x₂ - x₁), y₁ + k(y₂ - y₁)). An equivalent form is the weighted average P = ((b·x₁ + a·x₂)/(a + b), (b·y₁ + a·y₂)/(a + b)). Many teachers prefer the first form, because students can see the start point and the fraction of the way.
What is the difference between a ratio of 2:3 and 2/3 of the way?
They are different points. The ratio 2:3 compares the two pieces: PA is 2 parts and PB is 3 parts, 5 parts in all, so P is 2/5 of the way. "2/3 of the way" compares PA with the whole segment, which is the ratio 2:1. Confusing the two is a common error.
Why does the direction of the segment matter in HSG.GPE.B.6?
The ratio is measured from the starting point, so reversing the direction moves the point. On the segment from A(-3, 5) to B(5, -3), the ratio 1:3 from A gives (-1, 3), close to A, but the ratio 1:3 from B gives (3, -1), close to B. The word "directed" and the order of the letters tell you where to start.
How is the midpoint formula related to partitioning?
The midpoint partitions a segment in the ratio 1:1, so k = 1/2 and the partition formula becomes ((x₁ + x₂)/2, (y₁ + y₂)/2). For the midpoint the direction does not matter, which is one reason students are sometimes surprised that it matters for other ratios.
Why does the partition method work?
The segment from A to B is the hypotenuse of a slope triangle with legs Δx and Δy. The point P cuts off a smaller slope triangle at A that is similar to the whole triangle, with scale factor k = a/(a + b). So the horizontal and vertical legs are both multiplied by k, and P = (x₁ + kΔx, y₁ + kΔy). In the language of transformations, P is the image of B under the dilation centered at A with scale factor k.
What are common mistakes when partitioning a segment?
A common mistake is using a/b (such as 2/3 for the ratio 2:3) instead of a/(a + b). Others are:
starting from the wrong endpoint
losing the sign of Δx or Δy when the segment goes down or to the left
forgetting to add the starting coordinates after computing kΔx and kΔy
How can students check a partition point?
Three quick checks: the point must lie between the two endpoints; it must be closer to the start when a < b and closer to the end when a > b; and the distances PA and PB, found with the distance formula, must be in the ratio a:b. Students can also check that the slope from A to P equals the slope from A to B.
Where is partitioning a segment used later?
It is used with dilations and similarity (HSG.SRT.A.1), where a dilation centered at A with scale factor k sends B to the point k of the way from A. It returns in the section formula for vectors and in weighted averages, and computer graphics uses the same computation, called linear interpolation, to move smoothly from one point to another.
07
Related Standards
5 standards
These standards connect to HSG.GPE.B.6: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.RP.A.2Prerequisite
Recognize and represent proportional relationships between quantities
