In the figure below, the angle POQ is 90 degrees. What is the perimeter of the triangle OPQ ?

A 4\(\sqrt 2 \)

B 8\(\sqrt 2 \)

C 6 + 2\(\sqrt 5 \)

D 6

E 6\(\sqrt 2 \) + 2\(\sqrt {10} \)

Correct Answer: Option E

Solution:

অনুবাদঃ POQ একটি সমকোণী ত্রিভুজ । ত্রিভুজটির পরিসীমা বের করতে হবে ।

(x₁, y₁) এবং (x₂ , y₂)

দূরত্ব = \(\sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2}} \)

PO = \(\sqrt {{{( - 4 - 0)}^2} + {{(4 - 0)}^2}} = \sqrt {16 + 16} = \sqrt {32} \)

= \(\sqrt {2 \times 16} = 4\sqrt 2 \)

QO = \(\sqrt {{{(2 - 0)}^2} + {{(2 - 0)}^2}} = \sqrt {{2^{2\;}} + {2^{2\;}}} \)

= \(\sqrt {2 + {2^{2\;}}} = 2\sqrt 2 \)

PO = \(\sqrt {{{( - 4 - 2)}^2} + {{(4 - 2)}^2}} = \sqrt {{{( - 6)}^2} + {{(2)}^2}} \)

= \(\sqrt {40} = = 2\sqrt {10} \)

Perimeter of OPQ will be = \((4\sqrt 2 + 2\sqrt 2 + 2\sqrt {10} )\)

= 6\(\sqrt 2 \) + 2\(\sqrt {10} \)