Question

Let $AB$ be a line segment with midpoint $C$, and $D$ be the midpoint of $AC$. Let $C_1$ be the circle with diameter $AB$, and $C_2$ be the circle with diameter $AC$. Let $E$ be a point on $C_1$ such that $EC$ is perpendicular to $AB$. Let $F$ be a point on $C_2$ such that $DF$ is perpendicular to $AB$, and $E$ and $F$ lie on opposite sides of $AB$. Then the value of $\sin \angle FEC$ is

• A. $\frac{1}{\sqrt{10}}$
• B. $\frac{2}{\sqrt{10}}$
• C. $\frac{1}{\sqrt{13}}$
• D. $\frac{2}{\sqrt{13}}$

Answer 1

The correct option is A $\frac{1}{\sqrt{10}}$


In $△FEG$
$\tan \theta =\frac{x/4}{x/4+x/2}=\frac{1}{1+2}=\frac{1}{3}$
$\sin \theta =\frac{1}{\sqrt{1^2+3^2}}=\frac{1}{\sqrt{10}}$