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?
Step-by-step solution
Idea: A quadrilateral with four equal sides is a rhombus; if its diagonals are also equal (or one angle is a right angle), it is a square. All the lengths come from the distance formula.
- Plot the points (see the figure): A is in Quadrant I, B in II, C in III and D in IV.½ mark
- Sides: AB = √((−1 − 2)2 + (2 − 1)2) = √(9 + 1) = √10; BC = √((−2 + 1)2 + (−1 − 2)2) = √(1 + 9) = √10; CD = √((1 + 2)2 + (−2 + 1)2) = √(9 + 1) = √10; DA = √((2 − 1)2 + (1 + 2)2) = √(1 + 9) = √10.1 mark
- Diagonals: AC = √((−2 − 2)2 + (−1 − 1)2) = √(16 + 4) = √20; BD = √((1 + 1)2 + (−2 − 2)2) = √(4 + 16) = √20.1 mark
- All four sides are equal and the two diagonals are equal, so ABCD is a square. (Equal sides alone give a rhombus; equal diagonals make every angle 90°. Check: AB2 + BC2 = 10 + 10 = 20 = AC2, so ∠B = 90°.)½ mark
- Area = side2 = (√10)2 = 10 square units.1 mark
Check: Box method: ABCD fits inside the 4 × 4 square from (−2, −2) to (2, 2), area 16. The four corner triangles each have legs 1 and 3, area 1.5 each, total 6. 16 − 6 = 10 ✓.
Answer to write in the exam
AB = √(32 + 12) = √10, BC = √(12 + 32) = √10
CD = √(32 + 12) = √10, DA = √(12 + 32) = √10
AC = √(42 + 22) = √20, BD = √(22 + 42) = √20
All sides equal and diagonals equal ⇒ ABCD is a square
∴ Area = (√10)2 = 10 square units
Common mistakes that cost marks
- Stopping after showing the four sides are equal. A rhombus also has four equal sides; you must also show a right angle or equal diagonals.
- Giving the area as √10 (the side) or 20 (the diagonal squared). Area = side2 = 10.
- Joining the points in the wrong order (A to C) and calling AC a side. Join A → B → C → D → A.
How this can come in the exam
The area of the square with vertices (0, 0), (4, 1), (3, 5) and (−1, 4) is
- 17 square units
- √17 square units
- 34 square units
- 16 square units
Show answer
(A) 17 square units
Side = √(42 + 12) = √17, so area = (√17)2 = 17.
Assertion (A): A quadrilateral with all four sides equal is always a square.
Reason (R): A square has all four sides equal.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(D) A is false but R is true.
A is false: a rhombus has four equal sides but need not have right angles. R is true.
Try one yourself
Show that P (0, 1), Q (2, 0), R (3, 2) and S (1, 3) form a square, and find its area.
Show answer
PQ = QR = RS = SP = √5 and PR = QS = √10, so it is a square. Area = (√5)2 = 5 square units.
More questions like this
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- Copy the figure and mark S and Q in your diagram. Mark any point P in Quadrant I and any point R in Quadrant III, and write down their coordinates.