(a) Show that the points (−3,3), (3,−3) and (33,33) 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.
Get the complete, step-by-step math solution for: "(a) Show that the points ( -3, 3), (3, -3) and (3√(3), 3√(3)) are the vertices of an equilateral triangle. OR (b) Prove that A(4, 3), B(6, 4), C(5, 6)...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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 (x2−x1)2+(y2−y1)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=5) and both diagonals are equal (AC=BD=10), the quadrilateral ABCD is a square.