# What is the perimeter of a triangle with corners at #(1 ,5 )#, #(6 , 2 )#, and #(2 ,7 )#?

Perimeter

Perimeter

compute lengths of sides a, b , c

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To find the perimeter of the triangle with corners at (1, 5), (6, 2), and (2, 7), you need to calculate the sum of the lengths of its three sides.

Using the distance formula (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}), where ((x_1, y_1)) and ((x_2, y_2)) are the coordinates of two points:

For side 1: ((1, 5)) to ((6, 2)) [d_1 = \sqrt{(6 - 1)^2 + (2 - 5)^2} = \sqrt{5^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}]

For side 2: ((6, 2)) to ((2, 7)) [d_2 = \sqrt{(2 - 6)^2 + (7 - 2)^2} = \sqrt{(-4)^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}]

For side 3: ((2, 7)) to ((1, 5)) [d_3 = \sqrt{(1 - 2)^2 + (5 - 7)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}]

The perimeter, (P), is the sum of these distances: [P = d_1 + d_2 + d_3 = \sqrt{34} + \sqrt{41} + \sqrt{5}]

Therefore, the perimeter of the triangle is (P = \sqrt{34} + \sqrt{41} + \sqrt{5}).

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