(a) Show that the points $( -3, 3)$, $(3, -3)$ and $(3\sqrt{3}, 3\sqrt{3})$ are the vertices of an equilateral triangle. OR (b) Prove that A(4, 3), B(6, 4), C(5, 6), D(3, 5) are the vertices of a square ABCD.

Step 1: Calculate the lengths of the sides

To prove that the given points form a square, we first need to calculate the lengths of all four sides using the distance formula, which is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. If all four sides are equal, it could be either a square or a rhombus.

Step 2: Calculate the lengths of the diagonals

Next, we calculate the lengths of the diagonals $AC$ and $BD$ using the same distance formula. If the diagonals are also equal in length, then the figure is a square. If the diagonals were not equal, it would be a rhombus.

Step 3: Conclusion

Since all four sides are equal ($AB = BC = CD = DA = \sqrt{5}$) and both diagonals are equal ($AC = BD = \sqrt{10}$), the quadrilateral $ABCD$ is a square.

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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