Chapter 1: Orienting Yourself: The Use of Coordinates

Exercise

Question

Question: 16

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Plot the points A (2, 1), B (-1, 2), C (-2, -1) and D (1, -2) in the coordinate plane. Is ABCD a square? Can you explain why? What is the area of this square?

Answer

1
Find the lengths of all four sides of quadrilateral ABCD using the distance formula.
AB = \sqrt{(-1 - 2)^2 + (2 - 1)^2} = \sqrt{(-3)^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}
BC = \sqrt{(-2 - (-1))^2 + (-1 - 2)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
CD = \sqrt{(1 - (-2))^2 + (-2 - (-1))^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}
DA = \sqrt{(2 - 1)^2 + (1 - (-2))^2} = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}
2
Find the lengths of the two diagonals, AC and BD, using the distance formula.
AC = \sqrt{(-2 - 2)^2 + (-1 - 1)^2} = \sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20}
BD = \sqrt{(1 - (-1))^2 + (-2 - 2)^2} = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20}
3
Check whether all four sides are equal and whether the two diagonals are equal to determine whether ABCD is a square.
Answer: ABCD is a square.
\textbf{Reason: All four sides of ABCD are equal, with each side measuring $\sqrt{10}$, and the two diagonals are also equal, with each measuring $\sqrt{20}$. Therefore, ABCD is a square.}
Coordinate plane showing points A, B, C, and D forming a square.
Quadrilateral ABCD is a square.
4
Use the side length of the square to calculate its area.
Answer: Area of square ABCD = 10 square units.
\textbf{Reason: The side length is $\sqrt{10}$. Therefore, Area = $(side)^2 = (\sqrt{10})^2 = 10$. So, the area of ABCD is 10 square units.}
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