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8.G.B.8Common CoreMathGeometryGrade 8

8.G.B.8: Finding the Distance Between Two Points with the Pythagorean Theorem

In plain English: 8.G.B.8 is the Common Core grade 8 math standard that asks students to use the Pythagorean Theorem to find the distance between two points on a coordinate grid. Students draw a right triangle whose legs are the horizontal and vertical distances between the points, then find its hypotenuse. It is taught in Grade 8 Math and leads to the distance formula in high school.

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Understand and apply the Pythagorean Theorem.
Also written as 8.G.8 · Official standard

01

Lesson Plan

65-70 min

Overview

Students already know how to find the distance between two points that lie on the same horizontal or vertical line: subtract the coordinates that differ (6.NS.C.8). This lesson handles every other pair of points. The segment between two such points is slanted, so students draw a right triangle (a triangle with one 90° angle) under it: a horizontal leg, a vertical leg, and the segment itself as the hypotenuse, the side across from the right angle. The horizontal leg, the run, is the difference of the x-coordinates, and the vertical leg, the rise, is the difference of the y-coordinates. The Pythagorean Theorem then gives the distance: d² = run² + rise².

Points with negative coordinates are included from the start, since that is where many errors happen: from x = -4 to x = 1 is 5 units, not 3. Students also apply the idea to maps with a scale, where it gives the straight-line distance, and to the side lengths of triangles drawn on a grid. A distance is a length, so it is never negative, and it does not depend on which point you start from.

Learning Objectives

By the end of this lesson, students will be able to:

  • Draw the right triangle whose hypotenuse joins two given points on a coordinate grid
  • Find the lengths of the horizontal and vertical legs from the coordinates, including negative coordinates
  • Use the Pythagorean Theorem to find the distance between two points, as an exact square root and rounded to the nearest tenth
  • Find straight-line distances on scaled maps and side lengths of figures on a grid, and interpret them in context

Prior Knowledge Required

Students should already be comfortable with:

  • Using the Pythagorean Theorem to find the hypotenuse of a right triangle 8.G.B.7
  • Plotting points in all four quadrants and finding the distance between points with the same x-coordinate or the same y-coordinate 6.NS.C.8
  • Absolute value: the distance of a number from 0 on the number line, so |-4| = 4 6.NS.C.7
  • Square roots of perfect squares and estimates of other square roots (8.EE.A.2, 8.NS.A.2)

Lesson Procedure

65-70 minutes of class time across 5 phases.

  1. Warm-Up10 minutes

    Students plot three points on grid paper: A(1, 1), B(5, 1) and C(5, 4).

    Warm-Up Prompt

    "How long is segment AB? How long is segment BC? You can count grid squares for both. Now, how long is segment AC? Why can't you count squares for that one? What kind of triangle is ABC?"

    AB = 5 - 1 = 4 and BC = 4 - 1 = 3, by counting or subtracting. AC is slanted, so it cuts through squares diagonally and counting does not work. But ABC is a right triangle with its right angle at B, so AC is the hypotenuse: AC² = 4² + 3² = 25 and AC = 5. Tell students that every slanted segment on a grid can be handled this way.

  2. Direct Instruction20 minutes

    Model each step on a projected grid while students copy the steps into their notes.

    1. Plot the two points and join them with a segment. If they share an x-coordinate or a y-coordinate, the distance is the difference of the other coordinates, and you are done.
    2. Draw the right triangle: from one point go straight across (horizontally) and from the other straight up or down (vertically) until the two paths meet. The meeting point has the x-coordinate of one point and the y-coordinate of the other.
    3. Find the run: the horizontal leg is the difference of the x-coordinates. Subtract the smaller from the larger, so the length is positive. From -4 to 1 the run is 1 - (-4) = 5.
    4. Find the rise: the vertical leg is the difference of the y-coordinates, found the same way.
    5. Use the Pythagorean Theorem: the segment between the points is the hypotenuse, so d² = run² + rise² and d = √(run² + rise²).
    6. Give the answer as an exact square root, such as √29, and as a decimal rounded to the nearest tenth. On a map, multiply by the scale to get a real distance.
    • Points in different quadrants (Diagram 1)

      Find the distance between P(-4, -3) and Q(1, 9).

      Equation: The corner of the right triangle is (1, -3). Run: 1 - (-4) = 5. Rise: 9 - (-3) = 12. d² = 5² + 12² = 25 + 144 = 169, so d = √169 = 13 units.

    • A distance that is not a whole number

      Find the distance between A(1, 2) and B(6, 4).

      Equation: Run: 6 - 1 = 5. Rise: 4 - 2 = 2. d² = 25 + 4 = 29, so d = √29 ≈ 5.4 units. Estimate first: √29 is between 5 and 6, since 5² = 25 and 6² = 36.

    • Straight-line distance on a map (Diagram 2)

      On a town map, each block is 100 m. Home is at (-3, 2) and school is at (4, -2). How far is school from home in a straight line, and how far is it along the streets?

      Equation: Run: 4 - (-3) = 7 blocks. Rise: 2 - (-2) = 4 blocks. Straight line: √(49 + 16) = √65 ≈ 8.06 blocks, about 806 m. Along the streets: 7 + 4 = 11 blocks = 1,100 m. The straight line (for a drone or a bird) is about 294 m shorter.

    • Side lengths of a triangle on a grid

      A triangle has vertices (corners) J(-3, -1), K(1, 2) and L(4, -2). Find the length of each side. Are any two sides equal?

      Equation: JK: run 4, rise 3, so √(16 + 9) = 5. KL: run 3, rise 4, so √(9 + 16) = 5. JL: run 7, rise 1, so √(49 + 1) = √50 ≈ 7.1. JK = KL = 5, so the triangle has two equal sides.

    After Example 1, ask: "What if we had drawn the corner at (-4, 9) instead?" The triangle flips, but the legs are still 5 and 12, so the distance is the same. After Example 3, point out that the street path is the sum of the legs, and the straight line is the hypotenuse, which is always shorter.

  3. Guided Practice15 minutes

    Pairs solve each problem on grid paper. One partner plots and draws the triangle, the other writes the run, the rise and the equation; they switch after each problem.

    Guided practice problems with answers
    ProblemAnswer
    Find the distance between (0, 0) and (12, 35).Run 12, rise 35: √(144 + 1,225) = √1,369 = 37
    Find the distance between (-5, 2) and (7, -3).Run 7 - (-5) = 12, rise 2 - (-3) = 5: √169 = 13
    Find the distance between (2, -1) and (-3, 3), to the nearest tenth.Run 5, rise 4: √41 ≈ 6.4
    Find the distance between (-6, 4) and (5, 4). Do you need the Pythagorean Theorem?No: the points have the same y-coordinate, so the distance is 5 - (-6) = 11

    Listen for students who write 7 - 5 = 2 for the run in the second problem. Ask them to count the squares from -5 to 7 on the grid.

  4. Independent Practice15 minutes

    Students work alone, then compare with a partner. Each answer needs a sketch of the right triangle.

    Independent practice problems with answers
    ProblemAnswer
    Find the distance between (-6, -1) and (4, 23).Run 10, rise 24: √676 = 26
    Find the distance between (0, -4) and (20, 17).Run 20, rise 21: √841 = 29
    Find the distance between (3, -2) and (-3, 1), to the nearest tenth.Run 6, rise 3: √45 ≈ 6.7
    Sam says the distance between (-3, 2) and (9, 7) is √(6² + 5²). Find his mistake and the correct distance.He used 9 - 3 = 6 for the run. The run is 9 - (-3) = 12, so the distance is √(144 + 25) = 13
    On a school map drawn in feet, the gym door is at (120, 80) and the cafeteria door is at (300, 320). How far apart are they in a straight line?Run 180 ft, rise 240 ft: √90,000 = 300 ft
  5. Closure5-10 minutes

    Exit ticket: (1) Find the distance between (-2, -5) and (2, 3), exactly and to the nearest tenth. (Run 4, rise 8, √80 ≈ 8.9.) (2) Explain why a distance is never negative, even when the coordinates are. (3) Sketch the right triangle you would use for the points (-1, 6) and (3, -2), and label its legs.

Differentiation Strategies

For Struggling Students

  • Give grid paper with the points already plotted, and have students count the run and the rise square by square before subtracting
  • Start with points in the first quadrant only, then move one point into another quadrant
  • Use a fill-in frame: "run = ___ - ___ = ___, rise = ___ - ___ = ___, d = √(___² + ___²)"

For Advanced Students

  • Ask: which points with integer coordinates (whole numbers that are positive, negative or zero) are exactly 5 units from (0, 0)? Students should find 12 of them
  • Have students find a point (x, 4) that is 10 units from (1, -2) and explain why there are two answers (x = 9 and x = -7)
  • Extension (beyond this standard, in high school): write the run and the rise with letters for two points (x₁, y₁) and (x₂, y₂), which gives the distance formula of HSG.GPE.B.7

Assessment Guidance

What to Look For

A complete answer shows the right triangle on the grid, the corner point, the run and the rise as positive lengths, and the equation d² = run² + rise² before the square root. Watch for students who subtract a negative coordinate incorrectly (writing 1 - 4 for 1 - (-4)), who add the run and the rise instead of their squares, who forget the square root, and who forget to multiply by the map scale.

02

Classroom Activities

3 Activities

1

Human Coordinate Plane

15 minGroups of 4

Tape x- and y-axes on a floor with square tiles, so each tile is 1 unit (about 1 ft). Two students stand on two points, a third predicts the distance between them with the Pythagorean Theorem, and the fourth measures it with a tape measure.

The Point Pairs

  • Pair 1: (0, 0) and (3, 4)
  • Pair 2: (-2, 1) and (4, -1)
  • Pair 3: (-4, -2) and (5, 1)
  • Pair 4: (1, -4) and (-4, 1)

Answer Key

  • Pair 1: √25 = 5 units
  • Pair 2: run 6, rise 2, √40 ≈ 6.3 units
  • Pair 3: run 9, rise 3, √90 ≈ 9.5 units
  • Pair 4: run 5, rise 5, √50 ≈ 7.1 units
  • With 1-ft tiles, measured distances are usually within a few inches of these

Discussion Questions

  • In Pair 4, both points have a coordinate of -4. Why is the distance not 0 or 8?
  • Walk from one point to the other along the tile lines. Why is that walk always at least as long as the tape measure?

Modification for Distance Learning

Students use an online graphing tool, plot each pair, draw the right triangle, and compare their answer with the tool's distance measure.

2

Treasure Map Distances

20 minPairs

Each pair gets a printed map of a state park drawn on a grid, with each grid square 100 m. The camp is at the origin, (0, 0). Pairs find straight-line distances between landmarks and use them to plan a route.

The Landmarks

  • Camp (0, 0)
  • Lake (6, 8)
  • Cave (-5, 12)
  • Tower (9, -4)
  • Bridge (-7, -3)

Tasks

  • Find the distance from the camp to each landmark, in grid units and in meters
  • Find the distance from the lake to the tower and from the cave to the bridge
  • Which landmark is closest to the camp, and which is farthest?

Answer Key

  • Camp to lake: √100 = 10 units, 1,000 m
  • Camp to cave: √169 = 13 units, 1,300 m
  • Camp to tower: √97 ≈ 9.8 units, about 985 m
  • Camp to bridge: √58 ≈ 7.6 units, about 762 m
  • Lake to tower: √153 ≈ 12.4 units, about 1,237 m
  • Cave to bridge: √229 ≈ 15.1 units, about 1,513 m
  • Closest: the bridge. Farthest: the cave

Discussion Questions

  • The tower is 9 units to the right of the camp, farther across than the lake. Why is it still closer to the camp than the lake?
  • A hiker walks from the camp to the lake along the grid lines. How much farther is that than the straight line?
3

Equal Sides or Not?

20 minGroups of 3

Each group gets 4 triangle cards. For each triangle, students plot it, find all three side lengths, and sort it as isosceles (at least two equal sides) or scalene (no equal sides).

The 4 Triangle Cards

  • T1: (0, 0), (6, 0), (2, 5)
  • T2: (-3, -1), (3, -1), (3, 7)
  • T3: (-1, -3), (2, 3), (5, -3)
  • T4: (-2, 4), (3, 4), (0, 0)

Answer Key

  • T1: 6, √41 ≈ 6.4, √29 ≈ 5.4, scalene
  • T2: 6, 8, 10, scalene
  • T3: √45 ≈ 6.7, √45 ≈ 6.7, 6, isosceles
  • T4: 5, 5, √20 ≈ 4.5, isosceles

Discussion Questions

  • Only T2 has three whole-number side lengths. What is special about how T2 sits on the grid?
  • In T3, the two equal sides have the same run and the same rise. Is that the only way two sides can be equal? Look at T4.
  • Two sides of T1 are close in length. Why is it better to compare the squares (36 and 41) than the rounded decimals?

Challenge Variation

Groups design their own isosceles triangle with one horizontal side and trade it with another group to check.

03

Diagrams & Visual Aids

2 diagrams

Diagram 1: The Right Triangle Between Two Points

-5 -4 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x y P(-4, -3) Q(1, 9) (1, -3) run 5 rise 12 run: 1 - (-4) = 5 rise: 9 - (-3) = 12 d² = 5² + 12² d² = 25 + 144 = 169 d = 13 units
P(-4, -3) and Q(1, 9) on a square grid, drawn to scale. The right triangle has its corner at (1, -3). The run is 1 - (-4) = 5 and the rise is 9 - (-3) = 12, so PQ = √(5² + 12²) = 13 units.

Diagram 2: Straight Line or Along the Streets?

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 2 3 4 Home (-3, 2) School (4, -2) 7 blocks 4 blocks 1 block = 100 m Straight line: √(7² + 4²) = √65 ≈ 8.06 blocks ≈ 806 m Along the streets: 7 + 4 = 11 blocks = 1,100 m
A town map where each block is 100 m, drawn to scale. Home is at (-3, 2) and school at (4, -2). Along the streets (dashed) the trip is 7 + 4 = 11 blocks, or 1,100 m. The straight line is the hypotenuse, √65 ≈ 8.06 blocks, or about 806 m.

04

Homework Assignment

~30 min

8.G.B.8 Homework: Distance on the Coordinate Plane

Directions: Plot the points on grid paper and draw the right triangle for each distance. Show the run, the rise and the equation. Give exact answers as square roots and round to the nearest tenth.

Part 1: Distance Between Two Points (Problems 1-3)

  1. (a) Find the distance between (2, 3) and (26, 35). (b) Find the distance between (-4, 5) and (-4, -7). Which part did not need the Pythagorean Theorem, and why?
  2. Find the distance between (-6, -2) and (3, 4), exactly and to the nearest tenth. Then check that you get the same answer if you start from the other point.
  3. Jordan finds the distance between (-6, 1) and (3, 8). He writes run = 3 - 6 = -3, so d = √((-3)² + 7²) = √58. (a) What is his mistake? (b) Find the correct distance.

Part 2: Maps and Figures (Problems 4-6)

  1. On a city map, each grid unit is 0.2 mile. The stadium is at (-4, 3) and the train station is at (6, -3). (a) How far apart are they in a straight line, in miles? (b) How far is the trip along the streets, which follow the grid lines?
  2. A triangle has vertices A(-4, -1), B(4, 11) and C(12, -1). (a) Find the length of each side. (b) Is the triangle isosceles? Explain.
  3. In a video game, a character at (-3, -4) can reach any point 6 units away or less. Which of these points can it reach: (2, -1), (1, 2) and (-3, 2)? Show the distance to each.

Rubric

CriterionFull Credit (2 pts)Partial Credit (1 pt)No Credit (0 pts)
Right TriangleDrawn with the corner point and both legs labeledDrawn, but a leg or the corner is missingNo triangle
Run and RiseBoth legs correct, including across negative coordinatesOne leg wrongBoth legs wrong
DistanceCorrect exact answer and rounding, with units or the map scaleOne computing or rounding errorMissing or wrong method
ExplanationsClear reasons in Problems 1, 3 and 5Some reasons unclearNo reasons

05

Quiz: 20 Questions

Interactive, with answers

Instructions

Sketch the points on scratch paper for each question. Use a calculator where you need one, and round to the nearest tenth when an answer is not a whole number. 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

  1. Question 1 of 20 · Multiple Choice

    To find the distance between (2, 1) and (5, 8), you draw a right triangle with a horizontal leg and a vertical leg. How long are the legs?

  2. Question 2 of 20 · Multiple Choice

    What is the distance between (-1, 3) and (7, -2), to the nearest tenth?

  3. Question 3 of 20 · Multiple Choice

    What is the distance between (0, 0) and (9, 11), to the nearest tenth?

  4. Question 4 of 20 · Multiple Choice

    What is the distance between (-3, -2) and (4, 3), to the nearest tenth?

  5. Question 5 of 20 · Multiple Choice

    What is the distance between (-7, -2) and (5, -2)?

  6. Question 6 of 20 · Multiple Choice

    On a map, each grid unit is 50 m. The library is at (-2, 1) and the park is at (5, 9). To the nearest meter, how far apart are they in a straight line?

  7. Question 7 of 20 · Multiple Choice

    Which pair of points is farthest apart?

  8. Question 8 of 20 · Multiple Choice

    Two points have x-coordinates -6 and 2. How long is the horizontal leg of the right triangle between them?

  9. Question 9 of 20 · Multiple Choice

    A triangle has vertices (0, 0), (7, 0) and (2, 6). How long is its longest side, to the nearest tenth?

  10. Question 10 of 20 · Multiple Choice

    Which point is exactly 5 units from (2, -1)?

  11. Question 11 of 20 · Multiple Choice

    On a phone screen, positions are measured in pixels. An icon is at (100, 50) and a button is at (380, 260). How far apart are they?

  12. Question 12 of 20 · Multiple Choice

    Lena finds the distance between (-5, 1) and (2, 7). She writes √(3² + 6²) ≈ 6.7. What is the correct distance, to the nearest tenth?

  13. Question 13 of 20 · Multiple Choice

    On a hiking map, each grid unit is 0.5 km. The trailhead is at (-4, -1) and the summit is at (6, 4). How far is the summit from the trailhead in a straight line, to the nearest tenth?

  14. Question 14 of 20 · Multiple Choice

    Which expression gives the distance between (-1, 5) and (4, -4)?

  15. Question 15 of 20 · Short Answer

    Find the distance between (-8, -5) and (19, 31).

  16. Question 16 of 20 · Short Answer

    Find the distance between (2, -3) and (-8, 5), exactly and to the nearest tenth.

  17. Question 17 of 20 · Short Answer

    On a city map, each grid unit is 0.25 mile. A fire station is at (1, 2) and a fire is reported at (9, -1). How far is the fire from the station in a straight line, to the nearest tenth of a mile?

  18. Question 18 of 20 · Short Answer

    A triangle has vertices A(-2, 0), B(0, 7) and C(2, 0). Show that it is isosceles.

  19. Question 19 of 20 · Short Answer

    A rescue boat is at (-2, 3) on a harbor map. Buoy A is at (5, 14) and buoy B is at (10, -3). Which buoy is closer to the boat? Show both distances.

  20. Question 20 of 20 · Short Answer

    Explain why the distance between (-3, -3) and (3, 3) is not 6 + 6 = 12, and find the correct distance to the nearest tenth.

0 of 20 answered · 0 correct

06

Frequently Asked Questions

10 Questions

What does 8.G.B.8 mean?

8.G.B.8 means students use the Pythagorean Theorem to find how far apart two points on a coordinate grid are. They draw a right triangle with a horizontal leg and a vertical leg, find the legs from the coordinates, and compute the hypotenuse.

Do students need the distance formula for 8.G.B.8?

No. The standard asks students to apply the Pythagorean Theorem, so they draw the right triangle each time. The distance formula is the same idea written with letters, and it is usually taught in high school geometry (HSG.GPE.B.7). Students who understand the triangle will find the formula easy later.

How do you find the distance between two points with the Pythagorean Theorem?

Draw a right triangle with the segment between the points as the hypotenuse. The horizontal leg is the difference of the x-coordinates, and the vertical leg is the difference of the y-coordinates. Square both legs, add, and take the square root.

What do you do when a point has negative coordinates?

Subtract carefully, or count squares on the grid. The distance from -2 to 5 is 5 - (-2) = 7, because you cross 0. A common mistake is to write 5 - 2 = 3 instead.

What if the two points are on the same horizontal or vertical line?

Then there is no triangle: the distance is just the difference of the coordinates that change. For example, (2, -1) and (2, 6) are 7 units apart. The Pythagorean Theorem still works, with one leg equal to 0, but it is not needed.

Why is a distance never negative?

A distance is a length, and lengths are positive. When you subtract coordinates, take the larger minus the smaller, or use the absolute value. Squaring the legs also removes any negative sign, so the final distance is always positive.

Does it matter which point you start from?

No. Starting from the other point changes the signs of the differences, but not their lengths, so the legs and the distance are the same. Students can use this as a check.

How is 8.G.B.8 connected to slope?

Both use the same right triangle between two points. In 8.EE.B.6, rise ÷ run gives the slope of the line. In 8.G.B.8, √(run² + rise²) gives the length of the segment. One triangle answers both "how steep?" and "how long?"

Where is the distance between two points used in real life?

Maps and navigation apps use it for "as the crow flies" distances, video games use it to decide whether a character is in range, and robots and drones use it to plan straight paths. In each case, a grid of coordinates is laid over the space.

How can parents help with 8.G.B.8 at home?

Use a street map or a floor with square tiles. Pick two spots, count the blocks or tiles across and up, and ask your child to predict the straight-line distance, then measure it. Ask them to explain why the straight line is shorter than the walk along the grid.