How to Solve Pythagorean Theorem Problems? (+FREE Worksheet!)
In mathematics, the Pythagorean Theorem is the relationship between three sides of a right triangle.
Solve Pythagorean Theorem Problems: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Label the diagramWrite each given measurement on the figure.
- Choose the formulaMatch the formula to distance, midpoint, area, volume, or angle relationships.
- Check unitsUse linear, square, or cubic units as appropriate.
Worked examples
Rectangle area
- Area of a rectangle is length times width.
- Substitute 8 and 3.
- Use square units.
Midpoint
- Average the x-values.
- Average the y-values.
- Write the ordered pair.
Try one before moving on
Solve Pythagorean Theorem Problems: pop-up practice
Watch this practice video for additional examples and reinforcement:
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Step by step guide to solve Pythagorean Theorem problems
- We can use the Pythagorean Theorem to find a missing side in a right triangle.
- In any right triangle: \(\color{blue}{a^2+b^2= c^2}\)
For education statistics and research
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The Pythagorean Theorem – Example 1:
Right triangle ABC has two legs of lengths \(9\) cm (AB) and \(12\) cm (AC). What is the length of the third side (BC)?
Solution:
Use Pythagorean Theorem: \(\color{blue}{a^2+b^2= c^2}\)
Then: \(a^2+b^2= c^2 →9^2+12^2= c^2 →81+144=c^2\)
\(c^2=225 →\) \(c=\sqrt{225}=15\) \(\text{ cm }\) → \(c=15 \text{ cm }\)
The Pythagorean Theorem – Example 2:
Find the hypotenuse of the following right triangle.
Solution:
Use Pythagorean Theorem: \(\color{blue}{a^2+b^2= c^2}\)
Then: \(a^2+b^2= c^2 →8^2+6^2= c^2 →64+36=c^2\)
\(c^2=100 →\) \(c=\sqrt{100}=10\) → \(c=10\)
The Pythagorean Theorem – Example 3:
Find the hypotenuse of the following right triangle.
Solution:
Use Pythagorean Theorem: \(\color{blue}{a^2+b^2= c^2}\)
Then: \(a^2+b^2= c^2 →3^2+4^2= c^2 →9+16=c^2\)
\(c^2=25 →\) \(c=\sqrt{25}=5\) → \(c=5\)
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The Pythagorean Theorem – Example 4:
Right triangle ABC has two legs of lengths \(6\) cm (AB) and \(8\) cm (AC). What is the length of the third side (BC)?
Solution:
Use Pythagorean Theorem: \(\color{blue}{a^2+b^2= c^2}\)
Then: \(a^2+b^2= c^2 →6^2+8^2= c^2 →36+64=c^2\)
\( c^2=100 →\) \(c=\sqrt{100}=10\) \(\text{ cm }\) → \(c= 10 \text{ cm }\)
Exercises for Solveing the Pythagorean Theorem
Find the missing side in each right triangle.
Download Pythagorean Relationship Worksheet
- \(\color{blue}{13}\)
- \(\color{blue}{5}\)
- \(\color{blue}{15}\)
- \(\color{blue}{8}\)
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