The answer is: 13 units.
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Each side of the park is 13 units long.
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(Assuming hexagonal shape will have 6 (SIX) sides of equal length).
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Explanation:
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Let us assume you meant to write that the: "...new park, in the shape of hexagon, will have 6 (six) side of equal length."
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From the coordinates given, we can infer that this is a "regular" hexagon.
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Here is one way to solve the problem: Find the length of ONE side of the hexagon.
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Let us choose the following coordinates: (18,0), and (6.5, 5). Let the distance between these points , which would equal ONE side of our hexagon, represent "c", the hypotenuse of a right triangle. We want to solve for this value, "c".
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Let the distance on the x-axis, from (6.5, 0) to (18.5, 0); represent "b", one side of a right triangle.
→ We can solve for "b" ; → b = 18.5 - 6.5 ; → b = 12 .
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Let the distance from (6.5, 0) to (6.5, 5) ; represent "a"; the remaining side of the right triangle.
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→ a = y₂ - y₁ = 5 - 0 = 5 ;
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{Note: We choose the particular coordinates, including "(6.5, 0)", because the distances between the coordinates chosen form a "right triangle"; (with "c", representing a "hypotenuse", or "slanted line segment"; which would be also be "ONE line segment of the given regular hexagon", which is our answer, because each line segment is the same values, so we only have to find the value of ONE line segment, or side, of the hexagon.).
When considering the given coordinates: "(6.5, 5)", and "(18.5, 0)", a "right triangle" can be formed at the coordinate, "(6.5, 0),
By choosing this particular letters (variables) to represent the sides of a "right triangle", we can solve for the "hypotenuse, "c", using the Pythagorean theorem for the sides of a right triangle:
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→ a² + b² = c² ; in which "c" represents the hypotenuse of the right triangle
and "a" represents the length of one of the other sides; and "b" represents the length of the remaining side. (Note: All triangles have three sides).
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We have: a = 5 ; b = 12 ; → Solve for "c" ;
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→ a² + b² = c² ; ↔ c² = a² + b² ; Plug in the known values for "a" & "b" ;
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→ c² = a² + b² ; → c² = 5² + 12² ;
→ c² = 25 + 144 = 169 ; → c² = 169 ;
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→ Take the square root of each side; to isolate "c" on one side of the equation; and to solve for "c" ;
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→ c² = 169 ; √(c²) = √(169) ; → c = ± 13;
→ ignore the negative value; since the side of a polygon cannot be a negative number;
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→ c = 13 ; The answer is: 13 units.
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