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SubjectsMathematics
वर्गमूल सर्पिल
In this Class 9 Mathematics topic from Number Systems, students explore the square root spiral, a geometric construction that represents √2, √3, √4 and successive square-root lengths. They learn how right triangles and the Pythagorean theorem generate each new radius, connect these constructions with irrational numbers, and locate their values on the number line. The topic strengthens understanding of square roots, geometric representation, measurement, and the relationship between numerical patterns and visual reasoning.
TOPIC PRACTICE
Up to 5 questions from this page. Select your focus, then start.
Since \(5^2=25\) and \(6^2=36\), and \(25<32<36\), we get \(5<\sqrt{32}<6\). Therefore, the spiral distance \(\sqrt{32}\) lies between \(5\) and \(6\). The option between \(4\) and \(5\) is incorrect because \(\sqrt{32}>\sqrt{25}=5\). Exam tip: estimate a square root by locating the nearest perfect squares on either side of the number.
( \sqrt{17} ) is close to (4) because (17) is only (1) more than (16). Look for the number closest to a perfect square.
Consecutive radii are ( \sqrt{n} ), ( \sqrt{n+1} ), and ( \sqrt{n+2} ). Their squares are (n,n+1,n+2).
Since \(108=36\times3\), and \(36\) is the largest perfect-square factor of 108, \(\sqrt{108}=\sqrt{36}\times\sqrt{3}=6\sqrt{3}\). \(9\sqrt{2}\) is not correct because its square is \(81\times2=162\), not 108. Exam tip: first identify the largest perfect-square factor when simplifying a surd.
The direct answer is B: the distance is 11 units. The statement says that the spiral has been constructed up to sqrt{120}, and then sqrt{121} is constructed. Evaluate the new distance: 121=11f11, so sqrt{121}=11. Option A, 10, is not the new distance; 10 is sqrt{100}, an earlier perfect-square value. Option B, 11, is correct because its square is 121. Option C, 12, would have square 144, so it does not represent sqrt{121}. Option D, 121, is the number under the root, not the value of the square root. The key distinction is between the radicand, which is the number inside the root, and the square-root value, which is the distance represented. Thus the construction reaches a point 11 units from the starting point. Memory cue: ask, “What number multiplied by itself gives the number inside the root?” Since 11f11=121, the answer is 11.
QUIZ COMPLETE
