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The Pythagorean Theorem
Students explain and apply the Pythagorean Theorem and its converse to find lengths, classify triangles, and calculate coordinate distance.
Find It. Learn It. Master It.
Beginner
15–20 minutes
Triangles & Transformations
Triangles & Transformations
Page 7 of 8
Inside Cover
The Pythagorean Theorem
At a Glance
Binder SectionTriangles & Transformations
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
How does the Pythagorean Theorem connect right triangles to coordinate distance?
Build Understanding
In a right triangle, the legs a and b meet at the right angle and the hypotenuse c is opposite it. The Pythagorean Theorem states a squared + b squared = c squared.
To find a missing side, substitute known lengths, solve, and take the positive square root. For legs 6 and 8, c squared = 36 + 64 = 100, so c = 10.
The converse tests whether three side lengths form a right triangle. Put the longest length in place of c and compare. Equality means right; a squared + b squared greater than c squared means acute; less than means obtuse.
On a coordinate plane, horizontal and vertical changes form legs, producing the distance formula. Keep exact radical form when requested and round only at the end for an approximation.
Visual Model
4Thousands
3Hundreds
2Tens
7Ones
4,327 = 4,000 + 300 + 20 + 7
Goals
Learning Targets
- Identify legs and hypotenuse.
- Find a missing side in a right triangle.
- Explain a visual or algebraic proof.
- Use the converse to classify a triangle.
- Find coordinate distance.
- Simplify and approximate square roots appropriately.
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Interactive Vocabulary
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Reveal one step at a time
Worked Examples
Example 1
Legs 6 and 8
1
c²=36+64=100.
2
c=10.
Example 2
Distance from (−1,2) to (3,5)
1
Changes are 4 and 3.
2
d=√(4²+3²)=5.
Ask and explain
Common Questions
Can c be any side? No; c is the hypotenuse.
Why take only the positive root? Length cannot be negative.
Does the theorem work for every triangle? Only right triangles.
When should I round? At the final step unless told otherwise.
Notice and correct
Common Misconceptions
Using the hypotenuse as a leg
Locate the side opposite the right angle.
Forgetting the square root
c² is not c.
Subtracting coordinates inconsistently
Keep the same endpoint order; squares remove signs.
Learn Through Video
Watch, pause, and explain
Use the chapter markers, key vocabulary, and reflection prompts to stay actively engaged while you watch.
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Warm-Up
Guided
Independent
Challenge
1
Quick Start
Warm-Up
Find the hypotenuse with legs 5 and 12.
2
We Do
Guided Practice
Legs 9 and 12. Answer: hypotenuse 15.
Hypotenuse 13 and leg 5. Answer: other leg 12.
Do 8, 15, 17 form a right triangle? Answer: yes.
Classify 4, 5, 7. Answer: acute because 16 + 25 > 49.
Distance from (1,2) to (7,10). Answer: 10.
3
You Do
Independent Practice
Do 7,24,25 form a right triangle?
Find the diagonal of a 9 by 12 rectangle.
Find distance from (−2,−1) to (4,7).
4
Stretch Your Thinking
Challenge & Real-World Practice
Compare two routes across a rectangular park: along two sides and directly across. Calculate both, include units, and explain the practical tradeoff.
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Pythagorean Theorem: Find the Hypotenuse
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Open resource
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Before you begin, I can…
- I identify the hypotenuse before substituting.
- I square, combine, and take the principal square root.
- I estimate to check whether the length is reasonable.
1
Question 1
Which statement shows the key idea in this lesson?
2
Question 2
Which habit best supports accuracy?
3
Question 3
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4
Question 4
Which explanation is strongest?
5
Question 5
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