Determine if the vertices A(3,1), B(6,4), C(8,2), and D(5,−1) form a specific quadrilateral (like a rectangle, rhombus, or square), and find its perimeter.
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Step-by-Step Solution
Step 1: Calculate the lengths of the sides
To determine the type of quadrilateral, we first need to find the lengths of all its sides. We use the distance formula, which states that the distance between two points (x1,y1) and (x2,y2) is given by (x2−x1)2+(y2−y1)2. We apply this formula to find the lengths of sides AB, BC, CD, and DA.
Step 2: Calculate the lengths of the diagonals
Next, we calculate the lengths of the diagonals AC and BD using the same distance formula. This step is crucial to differentiate between a parallelogram, rectangle, rhombus, or square.
Step 3: Identify the type of quadrilateral
From the side lengths, we observe that AB=CD and BC=DA. This indicates that the opposite sides are equal, which means the quadrilateral is a parallelogram. Furthermore, since the diagonals AC=BD, a parallelogram with equal diagonals is a rectangle.
Step 4: Calculate the perimeter
The perimeter of a quadrilateral is the sum of the lengths of all its sides. We add the lengths of the four sides we calculated earlier to find the perimeter of the rectangle.