Grade 8 · Geometry · MAT-08-GEO-010
Find Distance in the Coordinate Plane
Represent and solve find distance in the coordinate plane problems using precise mathematical notation, multiple representations, and justified reasoning.
Learning objectives
What you should be able to do
- Represent and solve find distance in the coordinate plane problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify find distance in the coordinate plane results in context, including units, constraints, and reasonableness.
Prerequisite check
Make sure the foundation is ready.
Grade 8 uses this prerequisite as active knowledge in a more formal representation.
Learn
Build the idea from meaning, not memorization.
Meaning and structure
Find Distance in the Coordinate Plane belongs to geometry. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent find distance in the coordinate plane numerically and symbolically, then connect it to the Grade 8 Math Box. Tables, graphs, equations, coordinate models, diagrams, and verbal descriptions must agree.
Verification and interpretation
Check with substitution, an inverse operation, an equivalent representation, geometric invariants, estimation, or context. State whether the result is exact or approximate and preserve relevant units and constraints.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Pythagorean Model
- Model find distance in the coordinate plane in the Math Box.
- Change one input and predict which quantities or invariants should change.
- Explain what the visual, equation, and numerical result say about the same relationship.
Worked examples
Twenty different ways to see the concept work.
Find distance between (-5,-4) and (-2,0).
d = √(3² + 4²) ≈ 5Coordinate differences form the legs of a right triangle.
Find distance between (-4,-2) and (0,3).
d = √(4² + 5²) ≈ 6.4Coordinate differences form the legs of a right triangle.
Find distance between (-3,0) and (2,6).
d = √(5² + 6²) ≈ 7.81Coordinate differences form the legs of a right triangle.
Find distance between (-2,2) and (4,9).
d = √(6² + 7²) ≈ 9.22Coordinate differences form the legs of a right triangle.
Find distance between (-1,-3) and (6,5).
d = √(7² + 8²) ≈ 10.63Coordinate differences form the legs of a right triangle.
Find distance between (0,-1) and (3,3).
d = √(3² + 4²) ≈ 5Coordinate differences form the legs of a right triangle.
Find distance between (1,1) and (5,6).
d = √(4² + 5²) ≈ 6.4Coordinate differences form the legs of a right triangle.
Find distance between (2,-4) and (7,2).
d = √(5² + 6²) ≈ 7.81Coordinate differences form the legs of a right triangle.
Find distance between (-5,-2) and (1,5).
d = √(6² + 7²) ≈ 9.22Coordinate differences form the legs of a right triangle.
Find distance between (-4,0) and (3,8).
d = √(7² + 8²) ≈ 10.63Coordinate differences form the legs of a right triangle.
Find distance between (-3,2) and (0,6).
d = √(3² + 4²) ≈ 5Coordinate differences form the legs of a right triangle.
Find distance between (-2,-3) and (2,2).
d = √(4² + 5²) ≈ 6.4Coordinate differences form the legs of a right triangle.
Find distance between (-1,-1) and (4,5).
d = √(5² + 6²) ≈ 7.81Coordinate differences form the legs of a right triangle.
Find distance between (0,1) and (6,8).
d = √(6² + 7²) ≈ 9.22Coordinate differences form the legs of a right triangle.
Find distance between (1,-4) and (8,4).
d = √(7² + 8²) ≈ 10.63Coordinate differences form the legs of a right triangle.
Find distance between (2,-2) and (5,2).
d = √(3² + 4²) ≈ 5Coordinate differences form the legs of a right triangle.
Find distance between (-5,0) and (-1,5).
d = √(4² + 5²) ≈ 6.4Coordinate differences form the legs of a right triangle.
Find distance between (-4,2) and (1,8).
d = √(5² + 6²) ≈ 7.81Coordinate differences form the legs of a right triangle.
Find distance between (-3,-3) and (3,4).
d = √(6² + 7²) ≈ 9.22Coordinate differences form the legs of a right triangle.
Find distance between (-2,-1) and (5,7).
d = √(7² + 8²) ≈ 10.63Coordinate differences form the legs of a right triangle.
Practice
Guided practice with hints
01Find distance between (-1,1) and (2,5).
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: d = √(3² + 4²) ≈ 5
02Find distance between (0,-4) and (4,1).
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: d = √(4² + 5²) ≈ 6.4
03Find distance between (1,-2) and (6,4).
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: d = √(5² + 6²) ≈ 7.81
04Find distance between (2,0) and (8,7).
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: d = √(6² + 7²) ≈ 9.22
05Find distance between (-5,2) and (2,10).
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: d = √(7² + 8²) ≈ 10.63
06Find distance between (-4,-3) and (-1,1).
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: d = √(3² + 4²) ≈ 5
Practice
Independent practice
01Find distance between (-3,-1) and (1,4).
Answer: d = √(4² + 5²) ≈ 6.4
02Find distance between (-2,1) and (3,7).
Answer: d = √(5² + 6²) ≈ 7.81
03Find distance between (-1,-4) and (5,3).
Answer: d = √(6² + 7²) ≈ 9.22
04Find distance between (0,-2) and (7,6).
Answer: d = √(7² + 8²) ≈ 10.63
05Find distance between (1,0) and (4,4).
Answer: d = √(3² + 4²) ≈ 5
06Find distance between (2,2) and (6,7).
Answer: d = √(4² + 5²) ≈ 6.4
07Find distance between (-5,-3) and (0,3).
Answer: d = √(5² + 6²) ≈ 7.81
08Find distance between (-4,-1) and (2,6).
Answer: d = √(6² + 7²) ≈ 9.22
Common mistakes
Learn to catch the error, not just the answer.
Rigid transformations preserve size; dilations generally do not.
a²+b²=c² applies to right triangles.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Use transformations in design and coordinate geometry and Pythagorean reasoning for distance.
- Model capacities of cylindrical, conical, and spherical objects.
Challenge problems
Find distance between (-3,1) and (4,9).
d = √(7² + 8²) ≈ 10.63
Find distance between (-2,-4) and (1,0).
d = √(3² + 4²) ≈ 5
Find distance between (-1,-2) and (3,3).
d = √(4² + 5²) ≈ 6.4
Find distance between (0,0) and (5,6).
d = √(5² + 6²) ≈ 7.81
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve find distance in the coordinate plane problems and justify each result with a valid verification method.
Practice
Mastery check
01Find distance between (1,2) and (7,9).
Answer: d = √(6² + 7²) ≈ 9.22
02Find distance between (2,-3) and (9,5).
Answer: d = √(7² + 8²) ≈ 10.63
03Find distance between (-5,-1) and (-2,3).
Answer: d = √(3² + 4²) ≈ 5
04Find distance between (-4,1) and (0,6).
Answer: d = √(4² + 5²) ≈ 6.4
05Find distance between (-3,-4) and (2,2).
Answer: d = √(5² + 6²) ≈ 7.81
06Find distance between (-2,-2) and (4,5).
Answer: d = √(6² + 7²) ≈ 9.22
07Find distance between (-1,0) and (6,8).
Answer: d = √(7² + 8²) ≈ 10.63
08Find distance between (0,2) and (3,6).
Answer: d = √(3² + 4²) ≈ 5
Terminology
Words to know
- transformation
- A rule mapping a figure to an image.
- congruent
- Having the same size and shape.
- similar
- Having equal corresponding angles and proportional corresponding lengths.
- Pythagorean theorem
- For a right triangle, a²+b²=c².
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Grade 8 instructional guidance, mathematical practices, modeling, and progression toward high-school mathematics.Common Core State Standards for Mathematics — Grade 8
Grade-level expectations for real numbers, equations, functions, geometry, and statistics.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.