To find the perimeter of the rectangle defined by the vertices \((0, 0)\), \((4, 4)\), \((5, 3)\), and \((1, -1)\), we first need to determine if these points actually form a rectangle.
The coordinates provided do not suggest immediate rectangular symmetry, so we can verify whether the points form a rectangle by calculating the lengths of the sides and checking for right angles.
-
Calculate the distance between the points:
- Distance between \((0, 0)\) and \((4, 4)\): \[ d = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
- Distance between \((4, 4)\) and \((5, 3)\): \[ d = \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{1 + 1} = \sqrt{2} \]
- Distance between \((5, 3)\) and \((1, -1)\): \[ d = \sqrt{(1 - 5)^2 + (-1 - 3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
- Distance between \((1, -1)\) and \((0, 0)\): \[ d = \sqrt{(0 - 1)^2 + (0 + 1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
-
Side Lengths: From the calculations:
- Two sides are \(4\sqrt{2}\).
- Two sides are \(\sqrt{2}\).
-
Identify the Rectangle Structure: We confirm two pairs of equal sides (i.e., \(4\sqrt{2}\) and \(\sqrt{2}\)), consistent with the properties of a rectangle.
-
Calculate the Perimeter: The perimeter \(P\) of the rectangle can be calculated using the formula: \[ P = 2(\text{length} + \text{width}) = 2(4\sqrt{2} + \sqrt{2}) = 2(5\sqrt{2}) = 10\sqrt{2}. \]
-
Approximation to the Nearest Tenth: Using \(\sqrt{2} \approx 1.414\): \[ 10\sqrt{2} \approx 10 \times 1.414 = 14.14. \] Rounding to the nearest tenth gives: \[ \boxed{14.1}. \]
Thus, the perimeter of the rectangle is approximately \(14.1\) when rounded to the nearest tenth.
1 answer