Show that the points A(1,3),B(4,3),C(4,6) and D(1,6) are the vertices of a square.

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Calculate AB Distance: To determine if the given points form a square, we need to check two conditions: all sides are equal in length, and all angles are right angles. We will start by calculating the distances between consecutive points using the distance formula, which is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.Calculate BC Distance: Calculate the distance between points A(1,3) and B(4,3). $d_{AB} = \sqrt{(4 - 1)^2 + (3 - 3)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3$.Calculate CD Distance: Calculate the distance between points B(4,3) and C(4,6). $d_{BC} = \sqrt{(4 - 4)^2 + (6 - 3)^2} = \sqrt{0^2 + 3^2} = \sqrt{9} = 3$.Calculate DA Distance: Calculate the distance between points C(4,6) and D(1,6). $d_{CD} = \sqrt{(1 - 4)^2 + (6 - 6)^2} = \sqrt{(-3)^2 + 0^2} = \sqrt{9} = 3$.Confirm Equal Sides: Calculate the distance between points D(1,6) and A(1,3). $d_{DA} = \sqrt{(1 - 1)^2 + (6 - 3)^2} = \sqrt{0^2 + 3^2} = \sqrt{9} = 3$.Check Right Angles: Now that we have calculated the distances between consecutive points and found that $d_{AB} = d_{BC} = d_{CD} = d_{DA} = 3$, we can confirm that all sides are equal. The next step is to check the angles between the sides to ensure they are right angles.Calculate AB Slope: To check for right angles, we can use the slope formula, which is $m = \frac{y_2 - y_1}{x_2 - x_1}$. Perpendicular lines have slopes that are negative reciprocals of each other.Calculate BC Slope: Calculate the slope of line AB. Since A and B have the same y-coordinate, the slope is $m_{AB} = \frac{3 - 3}{4 - 1} = \frac{0}{3} = 0$, which means AB is a horizontal line.Confirm Right Angle: Calculate the slope of line BC. Since B and C have the same x-coordinate, the slope is $m_{BC} = \frac{6 - 3}{4 - 4} = \frac{3}{0}$, which is undefined, meaning BC is a vertical line.Confirm Square: Since AB is horizontal and BC is vertical, the angle between them is a right angle. We can assume the same for the other angles because the sides are all equal, and we have a pair of parallel lines (AB is parallel to CD, and BC is parallel to DA).Having equal sides and right angles at each vertex confirms that the quadrilateral ABCD is a square.

Show that the points $A(1,3), B(4,3), C(4,6)$ and $D(1,6)$ are the vertices of a square.

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Show that the points $A(1,3), B(4,3), C(4,6)$ and $D(1,6)$ are the vertices of a square.