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Pythagorean Theorem and Distance
Use precise geometric language, diagrams, and reasoning to use the Pythagorean Theorem to find lengths and distances.
Find It. Learn It. Master It.
Beginner
15–20 minutes
Geometry
Geometry
Page 20 of 21
Inside Cover
Pythagorean Theorem and Distance
At a Glance
Binder SectionGeometry
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
How can a labeled geometric model help us use the Pythagorean Theorem to find lengths and distances?
Visual Model
4Thousands
3Hundreds
2Tens
7Ones
4,327 = 4,000 + 300 + 20 + 7
Goals
Learning Targets
- I can name and label the important parts of a geometric figure.
- I can choose a definition, property, transformation, or formula that fits.
- I can use a diagram to explain and check my result.
Click each word
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.
Notice and correct
Common Misconceptions
Using a picture that does not match the question
Label every given length, angle, point, and shape.
Choosing a rule before identifying the figure
Name the shape and what is known first.
Mixing area, surface area, and volume
Use square units for area and cubic units for volume.
Assuming a drawing is to scale
Use the stated measurements and markings, not appearance alone.
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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Practice until you can explain it
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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
Create and solve your own geometry problem. Make every part of the visual match the question. || Create a new problem that asks someone to use the Pythagorean Theorem to find lengths and distances. Include a precise labeled diagram, solve it, and explain how the picture proves the result. | Answers vary | Identify the given geometric information and what you need to find. | Draw or label a matching figure before choosing a rule. | Use the labeled figure, apply the correct definition or formula, solve step by step, include units when needed, and check the result.
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Before you begin, I can…
- I can use correct geometric vocabulary.
- I can make a diagram that matches every part of a problem.
- I can justify a classification, measurement, or transformation.
1
Question 1
What should a useful geometry diagram include?
2
Question 2
What do congruent figures have?
3
Question 3
Which units describe area?
4
Question 4
What should you do before selecting a formula?
5
Question 5
How can a diagram help check an answer?
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