A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
A13
B17
C√119
D60
Apply a² + b² = c²: 5² + 12² = 25 + 144 = 169 = 13². So c = 13. This is the classic 5-12-13 Pythagorean triple. Note that 17 is a common distractor (5 + 12), and 60 is the product — neither is correct.
Question 2 True / False
The Pythagorean theorem states that for any triangle, a² + b² = c², where c is the longest side.
TTrue
FFalse
Answer: False
The theorem applies only to right triangles, and c must specifically be the hypotenuse — the side opposite the right angle. For a non-right triangle with sides 4, 5, 6, the equation 4² + 5² = 6² gives 41 ≠ 36, which is false but does not contradict the theorem (it just confirms the triangle is not a right triangle). The generalization to arbitrary triangles requires the law of cosines.
Question 3 Short Answer
How does the Pythagorean theorem lead directly to the distance formula between two points in the coordinate plane?
Think about your answer, then reveal below.
Model answer: The horizontal and vertical distances between the two points form the legs of a right triangle, and the straight-line distance is the hypotenuse. Applying a² + b² = c² gives d = √((x₂ − x₁)² + (y₂ − y₁)²).
This connection shows that the Pythagorean theorem is not just about triangles drawn on paper — it underlies the very definition of distance in the Cartesian plane. Every time you compute a distance in coordinate geometry, you are applying the Pythagorean theorem.