Question

If we join the points A and P, What will be the length of AP if BC = 2 cm.

• A. $2\sqrt{2+\sqrt{2}}$ cm
• B. $4\sqrt{4+\sqrt{2}}$ cm
• C. $2\sqrt{4+\sqrt{8}}$ cm
• D. $2\sqrt{1+\sqrt{2}}$ cm

Answer 1

The correct option is C

$2\sqrt{4+\sqrt{8}}$ cm

In the given figure :

PCB is a right-angled isosceles $△$ with right angle at C.

BC = PC = 2 cm

Using Pythagoras theorem in $△BPC$

PB = $\sqrt{2^2+2^2}$ = $\sqrt{8}$ cm

PB = AB , since they are radii of the same circle.

$\therefore$ AC = AB + BC = ($\sqrt{8}$ + 2) cm

Once again using Pythagoras theorem in $△APC$

AP = $\sqrt{A{C}^2+P{C}^2}$

AP = $\sqrt{(\sqrt{8}+2{)}^2+2^2}$

= $\sqrt{(8+4+4\sqrt{8})+4}$ (Using identity $(a+b{)}^2=a^2+2ab+b^2)$

= $\sqrt{16+4\sqrt{8}}$

= $\sqrt{4(4+\sqrt{8})}$

= $2\sqrt{4+\sqrt{8}}$ cm