Mathematics

Mathematics

Problem

On the same number line, obtain the positions of 13\sqrt{13} and 132\sqrt{\dfrac{13}{2}}. a. (i) Carry out a suitable construction to locate 13\sqrt{13} and mark it as point A. b. (ii) Without starting a fresh construction, use point A to obtain the point B representing 132\sqrt{\dfrac{13}{2}} on the same number line. c. (iii) Give a mathematical justification showing that the length OB (where O is the origin) is indeed 132\sqrt{\dfrac{13}{2}} units.
Intuitive explanation

The key is to use the geometric mean construction to locate square roots on a number line. For 13\sqrt{13}, find two numbers whose product is 13 (like 1 and 13), mark them on the line, then use a semicircle to drop a perpendicular and find where it hits the axis. For 13/2\sqrt{13/2}, apply the same method but with numbers whose product is 13/213/2 (like 1 and 6.56.5). Both constructions rely on the fact that the altitude to the hypotenuse of a right triangle equals the geometric mean of the segments it creates.

Intuitive explanation

The key is to use the geometric mean construction to locate square roots on a number line. For 13\sqrt{13}, find two numbers whose product is 13 (like 1 and 13), mark them on the line, then use a semicircle to drop a perpendicular and find where it hits the axis. For 13/2\sqrt{13/2}, apply the same method but with numbers whose product is 13/213/2 (like 1 and 6.56.5). Both constructions rely on the fact that the altitude to the hypotenuse of a right triangle equals the geometric mean of the segments it creates.

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