Chapter 4 · Lesson 9 of 12 · Grade 8 · Geometry · MAT-08-GEO-009
Find Missing Sides with the Pythagorean Theorem
Represent and solve find missing sides with the pythagorean theorem problems using precise mathematical notation, multiple representations, and justified reasoning.
Math V4 · Course chapter
Chapter 4: Transformations, Similarity, Pythagorean Geometry, and Volume
Geometry
Essential question: How can transformations and geometric relationships explain shape, distance, similarity, and measurement?
Where this lesson fits
Lesson 9 of 12. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Build and defend a multi-representation model using transformations, similarity, pythagorean geometry, and volume, including interpretation, verification, and error analysis.
Learning objectives
What you should be able to do
- Represent and solve find missing sides with the pythagorean theorem problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify find missing sides with the pythagorean theorem 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.
Chapter 4: Transformations, Similarity, Pythagorean Geometry, and Volume
Geometry. Lesson 9 of 12 extends earlier chapter reasoning.
Connection to the course
Use earlier Grade 8 chapters, Grade 7 prerequisite knowledge as active knowledge. Connect today's representation to at least one earlier representation and explain the relationship.
Meaning and structure
Find Missing Sides with the Pythagorean Theorem belongs to geometry. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent find missing sides with the pythagorean theorem 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 missing sides with the pythagorean theorem 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.
A right triangle has legs 3 and 4. Find the hypotenuse.
c = √(3² + 4²) = 5Apply a²+b²=c² and take the positive square root.
Represent the situation with a table, graph, diagram, number line, or equation before solving: A right triangle has legs 5 and 12. Find the hypotenuse.
c = √(5² + 12²) = 13A representation makes the mathematical structure visible before calculation. Apply a²+b²=c² and take the positive square root.
Explain why the method works, not only what steps to perform: A right triangle has legs 8 and 15. Find the hypotenuse.
c = √(8² + 15²) = 17Justification connects each step to a definition, property, or relationship. Apply a²+b²=c² and take the positive square root.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A right triangle has legs 7 and 24. Find the hypotenuse.
c = √(7² + 24²) = 25An estimate provides an independent reasonableness check. Apply a²+b²=c² and take the positive square root.
Verify the result with substitution, an inverse operation, or a second representation: A right triangle has legs 9 and 40. Find the hypotenuse.
c = √(9² + 40²) = 41Independent verification should support the result without merely repeating the same procedure. Apply a²+b²=c² and take the positive square root.
Interpret the result in context, including units, direction, rate, or geometric meaning: A right triangle has legs 6 and 8. Find the hypotenuse.
c = √(6² + 8²) = 10Interpretation connects a computed value to the mathematical situation. Apply a²+b²=c² and take the positive square root.
Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: A right triangle has legs 10 and 24. Find the hypotenuse.
Rigid transformations preserve size; dilations generally do not. Correct solution: c = √(10² + 24²) = 26Error analysis identifies the broken idea and replaces it with valid reasoning. Apply a²+b²=c² and take the positive square root.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: A right triangle has legs 16 and 30. Find the hypotenuse. Case B: A right triangle has legs 28 and 96. Find the hypotenuse.
Case A: c = √(16² + 30²) = 34 Case B: c = √(28² + 96²) = 100Comparison should address structure and representation, not only numerical answers. Apply a²+b²=c² and take the positive square root.
A right triangle has legs 14 and 48. Find the hypotenuse.
c = √(14² + 48²) = 50Apply a²+b²=c² and take the positive square root.
Represent the situation with a table, graph, diagram, number line, or equation before solving: A right triangle has legs 18 and 80. Find the hypotenuse.
c = √(18² + 80²) = 82A representation makes the mathematical structure visible before calculation. Apply a²+b²=c² and take the positive square root.
Explain why the method works, not only what steps to perform: A right triangle has legs 9 and 12. Find the hypotenuse.
c = √(9² + 12²) = 15Justification connects each step to a definition, property, or relationship. Apply a²+b²=c² and take the positive square root.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A right triangle has legs 15 and 36. Find the hypotenuse.
c = √(15² + 36²) = 39An estimate provides an independent reasonableness check. Apply a²+b²=c² and take the positive square root.
Verify the result with substitution, an inverse operation, or a second representation: A right triangle has legs 24 and 45. Find the hypotenuse.
c = √(24² + 45²) = 51Independent verification should support the result without merely repeating the same procedure. Apply a²+b²=c² and take the positive square root.
Interpret the result in context, including units, direction, rate, or geometric meaning: A right triangle has legs 21 and 72. Find the hypotenuse.
c = √(21² + 72²) = 75Interpretation connects a computed value to the mathematical situation. Apply a²+b²=c² and take the positive square root.
Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: A right triangle has legs 27 and 120. Find the hypotenuse.
Rigid transformations preserve size; dilations generally do not. Correct solution: c = √(27² + 120²) = 123Error analysis identifies the broken idea and replaces it with valid reasoning. Apply a²+b²=c² and take the positive square root.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: A right triangle has legs 12 and 16. Find the hypotenuse. Case B: A right triangle has legs 30 and 72. Find the hypotenuse.
Case A: c = √(12² + 16²) = 20 Case B: c = √(30² + 72²) = 78Comparison should address structure and representation, not only numerical answers. Apply a²+b²=c² and take the positive square root.
A right triangle has legs 20 and 48. Find the hypotenuse.
c = √(20² + 48²) = 52Apply a²+b²=c² and take the positive square root.
Represent the situation with a table, graph, diagram, number line, or equation before solving: A right triangle has legs 32 and 60. Find the hypotenuse.
c = √(32² + 60²) = 68A representation makes the mathematical structure visible before calculation. Apply a²+b²=c² and take the positive square root.
Explain why the method works, not only what steps to perform: A right triangle has legs 28 and 96. Find the hypotenuse.
c = √(28² + 96²) = 100Justification connects each step to a definition, property, or relationship. Apply a²+b²=c² and take the positive square root.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A right triangle has legs 36 and 160. Find the hypotenuse.
c = √(36² + 160²) = 164An estimate provides an independent reasonableness check. Apply a²+b²=c² and take the positive square root.
Practice
Guided practice with hints
01Verify the result with substitution, an inverse operation, or a second representation: A right triangle has legs 15 and 20. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: c = √(15² + 20²) = 25
02Interpret the result in context, including units, direction, rate, or geometric meaning: A right triangle has legs 25 and 60. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: c = √(25² + 60²) = 65
03Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: A right triangle has legs 40 and 75. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Rigid transformations preserve size; dilations generally do not. Correct solution: c = √(40² + 75²) = 85
04Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: A right triangle has legs 35 and 120. Find the hypotenuse. Case B: A right triangle has legs 63 and 280. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Case A: c = √(35² + 120²) = 125 Case B: c = √(63² + 280²) = 287
05A right triangle has legs 45 and 200. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: c = √(45² + 200²) = 205
06Represent the situation with a table, graph, diagram, number line, or equation before solving: A right triangle has legs 18 and 24. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: c = √(18² + 24²) = 30
Practice
Independent practice
01Explain why the method works, not only what steps to perform: A right triangle has legs 30 and 72. Find the hypotenuse.
Answer: c = √(30² + 72²) = 78
02Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A right triangle has legs 48 and 90. Find the hypotenuse.
Answer: c = √(48² + 90²) = 102
03Verify the result with substitution, an inverse operation, or a second representation: A right triangle has legs 42 and 144. Find the hypotenuse.
Answer: c = √(42² + 144²) = 150
04Interpret the result in context, including units, direction, rate, or geometric meaning: A right triangle has legs 54 and 240. Find the hypotenuse.
Answer: c = √(54² + 240²) = 246
05Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: A right triangle has legs 21 and 28. Find the hypotenuse.
Answer: Rigid transformations preserve size; dilations generally do not. Correct solution: c = √(21² + 28²) = 35
06Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: A right triangle has legs 35 and 84. Find the hypotenuse. Case B: A right triangle has legs 72 and 135. Find the hypotenuse.
Answer: Case A: c = √(35² + 84²) = 91 Case B: c = √(72² + 135²) = 153
07A right triangle has legs 56 and 105. Find the hypotenuse.
Answer: c = √(56² + 105²) = 119
08Represent the situation with a table, graph, diagram, number line, or equation before solving: A right triangle has legs 49 and 168. Find the hypotenuse.
Answer: c = √(49² + 168²) = 175
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
Explain why the method works, not only what steps to perform: A right triangle has legs 63 and 280. Find the hypotenuse.
c = √(63² + 280²) = 287
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A right triangle has legs 24 and 32. Find the hypotenuse.
c = √(24² + 32²) = 40
Verify the result with substitution, an inverse operation, or a second representation: A right triangle has legs 40 and 96. Find the hypotenuse.
c = √(40² + 96²) = 104
Interpret the result in context, including units, direction, rate, or geometric meaning: A right triangle has legs 64 and 120. Find the hypotenuse.
c = √(64² + 120²) = 136
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve find missing sides with the pythagorean theorem problems and justify each result with a valid verification method.
Practice
Mastery check
01Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: A right triangle has legs 56 and 192. Find the hypotenuse.
Answer: Rigid transformations preserve size; dilations generally do not. Correct solution: c = √(56² + 192²) = 200
02Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: A right triangle has legs 72 and 320. Find the hypotenuse. Case B: A right triangle has legs 33 and 44. Find the hypotenuse.
Answer: Case A: c = √(72² + 320²) = 328 Case B: c = √(33² + 44²) = 55
03A right triangle has legs 27 and 36. Find the hypotenuse.
Answer: c = √(27² + 36²) = 45
04Represent the situation with a table, graph, diagram, number line, or equation before solving: A right triangle has legs 45 and 108. Find the hypotenuse.
Answer: c = √(45² + 108²) = 117
05Explain why the method works, not only what steps to perform: A right triangle has legs 72 and 135. Find the hypotenuse.
Answer: c = √(72² + 135²) = 153
06Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A right triangle has legs 63 and 216. Find the hypotenuse.
Answer: c = √(63² + 216²) = 225
07Verify the result with substitution, an inverse operation, or a second representation: A right triangle has legs 81 and 360. Find the hypotenuse.
Answer: c = √(81² + 360²) = 369
08Interpret the result in context, including units, direction, rate, or geometric meaning: A right triangle has legs 30 and 40. Find the hypotenuse.
Answer: c = √(30² + 40²) = 50
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
Grade 8 scope, sequence, and public task structure
Coherent sequencing, representations, proof-oriented reasoning, and tasks.Eureka Math² Grade 8 program structure
Module coherence, cumulative practice, and algebraic fluency.enVision Mathematics Grade 8 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 8
Grade-level content and mathematical-practice expectations.