Write True or False and give reasons for your answer.
By geometrical construction,it is possible to divide a line segment in the ratio $\sqrt{3}: \frac{1}{\sqrt{3}}$.
By geometrical construction,it is possible to divide a line segment in the ratio $\sqrt{3}: \frac{1}{\sqrt{3}}$.
(A) True.
Given ratio $= \sqrt{3} : \frac{1}{\sqrt{3}}$.
To simplify this ratio,multiply both terms by $\sqrt{3}$:
$\sqrt{3} \times \sqrt{3} : \frac{1}{\sqrt{3}} \times \sqrt{3} = 3 : 1$.
Since $3 : 1$ is a ratio of two positive integers,it is always possible to divide a line segment in this ratio using a ruler and compass by constructing $3 + 1 = 4$ equal parts on an auxiliary ray.
Therefore,the statement is True.
Given ratio $= \sqrt{3} : \frac{1}{\sqrt{3}}$.
To simplify this ratio,multiply both terms by $\sqrt{3}$:
$\sqrt{3} \times \sqrt{3} : \frac{1}{\sqrt{3}} \times \sqrt{3} = 3 : 1$.
Since $3 : 1$ is a ratio of two positive integers,it is always possible to divide a line segment in this ratio using a ruler and compass by constructing $3 + 1 = 4$ equal parts on an auxiliary ray.
Therefore,the statement is True.
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