In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. 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
Solution:
We use the distance formula and basic concepts of coordinate geometry to solve the problem given.
To prove that the points A, B, C, and D from a square, the length of the four sides should be equal and the length of the two diagonals should be the same.
Let A (3, 4), B (6, 7), C (9, 4), and D (6, 1) be the positions of 4 friends.
We know that the distance between the two points is given by the distance formula: √{(x₁ - x₂)² + (y₁ - y₂)²}
Let's draw a structure to understand better as shown below.
To find AB, that is, the distance between points A (3, 4) and B (6, 7)
- x₁ = 3
- y₁ = 4
- x₂ = 6
- y₂ = 7
AB = √(3 - 6)² + (4 - 7)²
= √(- 3)² + (- 3)²
= √9 + 9
= √18
= 3√2
To find BC, that is, the distance between Points B (6, 7) and C (9, 4)
- x₁ = 6
- y₁ = 7
- x₂ = 9
- y₂ = 4
BC = √(6 - 9)² + (7 - 4)²
= √(- 3)² + (3)²
= √9 + 9
= √18
= 3√2
To find CD, that is, the distance between Points C (9, 4) and D (6, 1)
- x₁ = 9
- y₁ = 4
- x₂ = 6
- y₂ = 1
CB = √{(9 - 6)² + (4 - 1)²}
= √(3)² + (3)²
= √9 + 9
= √18
= 3√2
To find AD, that is, the distance between Points A (3, 4) and D (6, 1)
- x₁ = 3
- y₁ = 4
- x₂ = 6
- y₂ = 1
AD = √(3 - 6)² + (4 - 1)²
= √(- 3)² + (3)²
= √9 + 9
= √18
= 3√2
Now, let's find the length of the diagonals.
To find AC, that is, the distance between points A (3, 4) and C (9, 4)
- x₁ = 3
- y₁ = 4
- x₂ = 9
- y₂ = 4
Diagonal AC = √(3 - 9)² + (4 - 4)²
= √(- 6)² + 0²
= 6
To find BD, that is, the distance between Points B (6, 7) and D (6, 1)
- x₁ = 6
- y₁ = 7
- x₂ = 6
- y₂ = 1
Diagonal BD = √(6 - 6)² + (7 - 1)²
= √(0² + (6)²
= 6
The four sides AB, BC, CD, and AD are of the same length, and diagonals AC and BD are of equal length. Therefore, ABCD is a square and hence, Champa was correct
☛ Check: NCERT Solutions Class 10 Maths Chapter 7
Video Solution:
In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. 7.8. 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
NCERT Class 10 Maths Solutions Chapter 7 Exercise 7.1 Question 5
Summary:
If in a classroom, 4 friends are seated at points A, B, C, and D as shown in the above 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. Hence the four sides AB, BC, CD, and AD are of the same length, and diagonals AC and BD are of equal length. Therefore, ABCD is a square and hence, Champa was correct.
☛ Related Questions:
- Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:(i) (- 1, - 2), (1, 0), (- 1, 2), (- 3, 0) (ii) (- 3, 5), (3, 1), (0, 3), (- 1, - 4) (iii) (4, 5), (7, 6), (4, 3), (1, 2)
- Find the point on the x-axis which is equidistant from (2, - 5) and (- 2, 9).
- Find the values of y for which the distance between the points P (2, - 3) and Q (10, y) is 10 units.
- If Q (0, 1) is equidistant from P (5, - 3) and R (x, 6), find the values of x. Also, find the distances QR and PR.