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.

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 $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. 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.

More from this chapter

• Q7. If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b. [2 Marks]
• Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6).
• Find a point on the y-axis that is at the same distance from the points (-5, 2) and (9, -2).
• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
• 1. Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (-a, -b)
• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
• Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.

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