The Converse of the Pythagorean Theorem
Apply the Converse of the Pythagorean Theorem in Grade 10 geometry: if a²+b²=c², then the triangle is right; use it to classify triangles as acute, right, or obtuse.
Key Concepts
Property If the sum of the squares of the lengths of the two shorter sides of a triangle equals the square of the length of the longest side, then the triangle is a right triangle.
Do side lengths 8, 15, and 17 form a right triangle? Check: $8^2 + 15^2 \stackrel{?}{=} 17^2 \rightarrow 64 + 225 \stackrel{?}{=} 289 \rightarrow 289 = 289$. Yes, it is a right triangle. Do side lengths 10, 20, and 23 form a right triangle? Check: $10^2 + 20^2 \stackrel{?}{=} 23^2 \rightarrow 100 + 400 \stackrel{?}{=} 529 \rightarrow 500 \neq 529$. Nope, not a right triangle.
This is the detective version of the theorem! It lets you test if any triangle is secretly a right triangle. Just take the three side lengths, plug them into $a^2 + b^2 = c^2$, making sure 'c' is the longest side. If the equation is true, you've found a right triangle. If not, the triangle is just an imposter.
Common Questions
What does the Converse of the Pythagorean Theorem state?
If a²+b²=c² for sides a, b, c of a triangle (c being the longest), then the triangle is a right triangle with the right angle opposite side c.
Is a triangle with sides 5, 12, 13 a right triangle?
Check: 5²+12²=25+144=169=13². Since c²=a²+b², yes, it is a right triangle.
How do you classify a triangle with sides 4, 5, 7 as acute, right, or obtuse?
Compare a²+b² to c²: 4²+5²=41, 7²=49. Since 41<49 (a²+b²<c²), the triangle is obtuse.