Find the perimeter of the triangle formed by the points $(3, 5), (4, 8)$ and $(5, 6)$ .

\sqrt{5} (2 + \sqrt{2})

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\sqrt{3} (2 + \sqrt{2})

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\sqrt{2} (5 + \sqrt{3})

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\sqrt{5} (2 + \sqrt{3})

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Solution

The correct option is A $\sqrt{5} (2 + \sqrt{2})$
To find the perimeter of the triangle formed by the points $A (3, 5)$ , $B (4, 8)$ and $C (5, 6)$ , we must find the distance between these three points that can be determined as follows:

$A B = \sqrt{{(4 - 3)}^{2} + {(8 - 5)}^{2}} = \sqrt{1^{2} + 3^{2}} = \sqrt{1 + 9} = \sqrt{10}$

$B C = \sqrt{{(5 - 4)}^{2} + {(6 - 8)}^{2}} = \sqrt{1^{2} + 2^{2}} = \sqrt{1 + 4} = \sqrt{5}$

$A C = \sqrt{{(5 - 3)}^{2} + {(6 - 5)}^{2}} = \sqrt{2^{2} + 1^{2}} = \sqrt{4 + 1} = \sqrt{5}$

Now the perimeter is $A B + B C + A C$ that is

$A B + B C + A C = \sqrt{10} + \sqrt{5} + \sqrt{5} = \sqrt{10} + 2 \sqrt{5} = \sqrt{5} (2 + \sqrt{2})$

Hence the perimeter is $\sqrt{5} (2 + \sqrt{2})$ .

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