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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
A right triangle has legs of 6 units and 8 units. Find the length of the hypotenuse.
1
Find what you know — The legs are the two sides that make the right angle. They are 6 units and 8 units long. The missing side is the hypotenuse, the side across from the right angle. We will call it c.
2
Choose the rule — This is a right triangle, so use the Pythagorean Theorem: a²+b²=c². This rule says that the squares of the two legs add to the square of the hypotenuse.
3
Substitute the numbers — Substitute means replace each letter with the number it stands for. Replace a with 6 and b with 8: 6²+8²=c².
4
Square and add — A square means multiply a number by itself. So 6²=6×6=36 and 8²=8×8=64. Add: 36+64=100. Now we know c²=100.
5
Find c — The symbol c² means c×c. We need the number that multiplies by itself to make 100. Use the square root: c=√100=10.
6
Answer and check — The hypotenuse is 10 units long. Check the math: 6²+8²=36+64=100, and 10²=100. The answer makes sense because the hypotenuse should be longer than either leg.
Example 2
Distance from (−1,2) to (3,5)
1
Find the changes — Move 4 units across and 3 units up. These two changes make the legs of a right triangle.
2
Choose the rule — Use the distance formula, which comes from the Pythagorean Theorem.
3
Substitute and solve — d=√(4²+3²)=√(16+9)=√25=5 units.
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
Directions: Use a²+b²=c². Show the substitution and square-root step.
A right triangle has legs 5 units and 12 units. What is the length of the hypotenuse?
Picture the lengthUse the picture to organize what you know before you solve.
2
We Do
Guided Practice
Directions: Use a²+b²=c². Show the equation used to find the missing leg.
A right triangle has a hypotenuse of 10 units and one leg of 6 units. What is the length of the other leg?
Plan your solutionRecord what you know, choose a model or rule, and solve without revealing the final answer.
Step 1: Label the known hypotenuse, known leg, and unknown leg.
Step 2: Rearrange the theorem so the unknown leg is isolated before doing any arithmetic.
Directions: Use the distance formula. Show both coordinate differences before simplifying.
What is the distance between (0,0) and (8,15)?
Picture the distanceUse the picture as you work through each step.
Step 1: Find the horizontal change and vertical change without calculating the distance.
Step 2: Place those two changes into the distance formula, but stop before simplifying the radical.
3
You Do
Independent Practice
Directions: Use the converse of the Pythagorean Theorem. Compare the sum of the squares of the two shorter sides with the square of the longest side.
Do side lengths 7, 24, and 25 form a right triangle? Answer yes or no.
Picture the lengthUse the picture to organize what you know before you solve.
Directions: Use the Pythagorean Theorem. Show the equation and include units.
A rectangle is 9 units wide and 12 units long. What is the length of its diagonal?
Picture the lengthUse the picture to organize what you know before you solve.
Directions: Use the distance formula. Show the horizontal and vertical changes before simplifying.
What is the distance between (−2,−1) and (4,7)?
Picture the distanceUse the picture to organize what you know before you solve.
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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IXL
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?
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Question 3
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Question 4
Which explanation is strongest?
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