Question

A square with side length $1\text{cm}$ has an inscribed semicircle along one of its sides as shown in figure. The line $AE$ is tangent to the semicircle. What is the length of $AE$ ?

• A. $\frac{5}{6}\text{cm}$
• B. $1\text{cm}$
• C. $\frac{5}{4}\text{cm}$
• D. $\frac{6}{5}\text{cm}$

Answer 1

The correct option is C $\frac{5}{4}\text{cm}$

A pair of tangents from outside a circle have equal length.

Since, $AD$ and $AF$ are the tangents from the point $A$ to the semicricle. The length of $AD$ and $AF$ are equal by the principle of equal tangents.
$\Rightarrow AD=AF=1$
Let $EF=y$
$\therefore CE=y$ (by the principle of equal tangents)
$\Rightarrow BE=1-y$
In $\Delta ABE,$
$1^2+(1-y{)}^2=(1+y{)}^2$
$\Rightarrow y=\frac{1}{4}$
$\therefore AE=1+\frac{1}{4}=\frac{5}{4}\text{cm}$