Q) In a classroom, 4 friends are seated at the points A, B , C , and D as shown in Figure. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, Don’t you think ABCD is a square? Chameli disagrees. Using distance formula, find which of them is correct.

Ans:

To ensure if ABCD is a square or not, we need to ensure 2 points:

• a) Sides AB, BC, CD, AD should be equal
• b) Both diagonals, AC and BD, should be equal
• c) ∠A, ∠B, ∠C and ∠D should be 90⁰ i.e. diagonals AC and BD should satisfy Pythagoras theorem.

Step 1: We consider (0,0) as origin and find out coordinates of point A, B, C & D from the diagram:

Coordinates of point A: (3, 4)

Coordinates of point B: (6, 7)

Coordinates of point C: (9, 4)

Coordinates of point D: (6, 1)

Step 2: We calculate sides AB, BC, CD and AD:

We know that the distance between two points P (X₁, Y₁) and Q (X₂, Y₂) is given by:

PQ =

For AB, we have coordinates as A (3,4) and B (6, 7)

∴ AB =

∴ AB = = 3 √ 2 units

For BC, we have coordinates as B (6, 7) and C (9, 4)

∴ BC =

∴ BC = = 3 √ 2 units

For CD, we have coordinates as C (9,4) and D (6, 1)

∴ CD =

∴ CD = = 3 √ 2 units

For AD, we have coordinates as A (3, 4) and D (6, 1)

∴ AD =

∴ AD =  = 3 √ 2 units

Since, AB = BC = CD = AD, hence our 1st condition is satisfied.

Step 3: Let’s calculate the length of the diagonals AC and BD

For AC, we have coordinates as A (3,4) and C(9, 4)

∴ AC =

∴ AC = = 6 units

For BD, we have coordinates as B (6,7) and D(6, 1)

∴ BD =

∴ BD = = 6 units

Since, AC = BD, hence our 2nd condition is satisfied.

Step 4: in Δ ABC, if ∠B is 90⁰, then AB² + BC² = AC²

Let’s check this by taking values of AB and BC from step 2:

∴ LHS = (3√ 2)² + (3 √ 2 )²

= (18) + (18) = 36

= (6)² = AC² =  RHS   (from step 3: AC = 6 units)

Thus, ∠ B = 90⁰

Similarly, in Δ ADC, if ∠D is 90⁰, then AD² + CD² = AC²

Let’s check this by taking values of AD and CD from step 2:

∴ LHS = (3√ 2)² + (3 √ 2 )²

= (18) + (18) = 36

= (6)²

= AC² =  RHS   (from step 3: AC = 6 units)

Thus, ∠ D= 90⁰

Similarly, we prove that in Δ BAD, ∠ A = 90⁰

and in Δ BCD, ∠ C= 90⁰

Since, ∠A = ∠B = ∠C = ∠D = 90⁰, hence our 3rd condition is also satisfied.

Therefore, ABCD is a square, and hence Champa is right.

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