NCERT Class X Chapter 7: Coordinate Geometry Exercise 7.1 Question 5
NCERT Class X Chapter 7: Coordinate Geometry
Question:
In a classroom, 4 friends are seated at the points A, B, C and D as shown in the 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.
Given:
The coordinates of the points are:
- A(1, 1)
- B(4, 1)
- C(4, 5)
- D(1, 5)
To Find:
Using the distance formula, determine whether ABCD is a square.
Formula:
Distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
$$
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$
Solution:
Step 1: Find the length of side AB.
$$ AB = \sqrt{(4-1)^2 + (1-1)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3 $$Step 2: Find the length of side BC.
$$ BC = \sqrt{(4-4)^2 + (5-1)^2} = \sqrt{0^2 + 4^2} = \sqrt{16} = 4 $$Step 3: Find the length of side CD.
$$ CD = \sqrt{(1-4)^2 + (5-5)^2} = \sqrt{(-3)^2 + 0^2} = \sqrt{9} = 3 $$Step 4: Find the length of side DA.
$$ DA = \sqrt{(1-1)^2 + (1-5)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{16} = 4 $$Step 5: Find the lengths of the diagonals AC and BD.
$$ AC = \sqrt{(4-1)^2 + (5-1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$ $$ BD = \sqrt{(1-4)^2 + (5-1)^2} = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$Step 6: Analyze the side lengths and diagonals to check if ABCD is a square.
All sides of a square must be equal and both diagonals must also be equal.
Here, AB = CD = 3 units, BC = DA = 4 units, and both diagonals are 5 units.
Since not all four sides are equal, ABCD is not a square.
Result:
Chameli is correct. ABCD is not a square; it is a rectangle.
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