Question
If a right triangle has sides 5, 12, and 13 units, then a triangle with sides 10, 24, and 26 is also a right triangle.
True, because multiples of a right triangle triple produces a similar right triangle with congruent angles.
False, because doubling the sides of the triangle does not keep the angles congruent.
True, because multiples of a right triangle triple produces a similar right triangle with congruent angles.
False, because doubling the sides of the triangle does not keep the angles congruent.
Answers
GPT-4o mini
The correct answer is **True**.
In a right triangle, the relationship defined by the Pythagorean theorem (a² + b² = c²) holds. The sides of the triangle with lengths 5, 12, and 13 satisfy this condition, as:
5² + 12² = 25 + 144 = 169 = 13².
When you multiply the sides of a right triangle by the same factor, the new triangle retains its angles because it is similar to the original triangle. Thus, the triangle with sides 10, 24, and 26 is also a right triangle:
10² + 24² = 100 + 576 = 676 = 26².
So the angles are congruent, confirming that the statement is true. Multiples of a right triangle produce another right triangle with the same angles.
In a right triangle, the relationship defined by the Pythagorean theorem (a² + b² = c²) holds. The sides of the triangle with lengths 5, 12, and 13 satisfy this condition, as:
5² + 12² = 25 + 144 = 169 = 13².
When you multiply the sides of a right triangle by the same factor, the new triangle retains its angles because it is similar to the original triangle. Thus, the triangle with sides 10, 24, and 26 is also a right triangle:
10² + 24² = 100 + 576 = 676 = 26².
So the angles are congruent, confirming that the statement is true. Multiples of a right triangle produce another right triangle with the same angles.