In the proof of (\sqrt{3}), if (h=3r) and (k=3s) are proved, what happens to the lowest fraction condition?
Answer and explanation
Correct answer: It breaks because (3) is a common factor
The square-root spiral repeatedly forms a right triangle. At each stage, the old hypotenuse becomes one leg of the next right triangle, and a new perpendicular leg of length 1 is drawn. If the old hypotenuse is \(\sqrt{n}\), the new one is \(\sqrt{n+1}\), because of the Pythagorean theorem.
If \(h=3r\) and \(k=3s\), then both the numerator and denominator contain the common factor 3. A fraction in lowest form must have numerator and denominator with no common factor greater than 1. Thus the lowest-fraction condition is contradicted; it does not become stronger, make the root an integer, or prove \(k=0\). Therefore option C correctly describes the effect.
Frequently asked questions
What is the correct answer to this question?
It breaks because (3) is a common factor
Why is this the correct answer?
The square-root spiral repeatedly forms a right triangle. At each stage, the old hypotenuse becomes one leg of the next right triangle, and a new perpendicular leg of length 1 is drawn. If the old hypotenuse is \(\sqrt{n}\), the new one is \(\sqrt{n+1}\), because of the Pythagorean theorem.
If \(h=3r\) and \(k=3s\), then both the numerator and denominator contain the common factor 3. A fraction in lowest form must have numerator and denominator with no common factor greater than 1. Thus the lowest-fraction condition is contradicted; it does not become stronger, make the root an integer, or prove \(k=0\). Therefore option C correctly describes the effect.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Proof of irrationality of square root 2 and square root 3.
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