Question

By geometrical construction it is possible to divide a line segment in the ratio 2 + root 3 ratio 2- root 3.


True or False?????

Answer 1

Answer:

False.

As 2+√3 : 2-√3 can be simplified as 7+4√3 : 7+4√3 is not a positive integee, while 1 is.

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Answer 2

Given,

For geometrical construction, a line segment is divided into ratios 2+√3 and 2-√3.

To Find,

Whether it is possible to construct this.

Solution,

By the rule, we can say that, If the ratio is in form of any positive integer then any line segment can be divided in that ratio.

Here given ratio is,

2 + √3 : 2 - √3

To simplify this we can say this ratio =

$\frac{(2+\sqrt{3}) }{(2-\sqrt{3})}$

[Now let's Multiply and divide the upper side and lower side by (2 + √3)]

= $\frac{2+\sqrt{3}}{2-\sqrt{3}} .\frac{2+\sqrt{3}}{2+\sqrt{3}}.$

= $\frac{4+3+2.2.\sqrt{3}}{4-3}$.

=$\frac{7+4\sqrt{3}}{1}$

So we get the ratio

(2 + √3 ): (2 - √3)=(7 + 4√3) : 1.

We get here 1 is a positive integer but (7 + 4√3) is not an integer.

So It is not possible to divide a line segment with the ratio,  (2 + √3 ): (2 - √3)=(7 + 4√3): 1.

Hence, the statement  is False.