8.G.B.7
8th Grade
Geometry
Pythagorean Theorem
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
How to explain it
The anchor students hold onto: a² + b² = c². To find c (hypotenuse): square the legs, add, take √. To find a missing leg: subtract the known squared values, then take √. Round with ≈ if irrational.
Pythagorean Theorem side lengths power #97 Distance on the Coordinate Plane and extend into 3D diagonal problems in high school geometry and Algebra 1.
Worked examples
Example 1
Legs 6, 8. Find hypotenuse c.
Step 1Substitute: 6² + 8² = c².
Step 236 + 64 = 100 = c².
Step 3c = √100.
Answerc = 10.
Example 2
Hyp 13, one leg 5. Find leg.
Step 1Rearrange: leg = √(c² − known²).
Step 2√(13² − 5²) = √(169 − 25).
Step 3√144.
AnswerMissing leg = 12.
Example 3
Legs 3, 5. Find c, nearest 0.1.
Step 13² + 5² = c².
Step 29 + 25 = 34 = c².
Step 3c = √34.
Answerc ≈ 5.8.
Common mistakes
What students write
Adding legs before squaring: writing (6 + 8)² = c² instead of 6² + 8² = c².
The fix
Square each leg first, then add. The formula is a² + b² = c², not (a + b)² = c².
What students write
Assigning c to a leg instead of the hypotenuse. Only the side opposite the right angle is c.
The fix
Find the right angle first (∠ = 90°). The side directly across from it is always the hypotenuse c.
Teacher tip
Head off the two predictable errors before they happen. First: Square each leg first, then add. The formula is a² + b² = c², not (a + b)² = c². Second: Find the right angle first (∠ = 90°). The side directly across from it is always the hypotenuse c.