Ex.7.1 Q5 Coordinate Geometry Solution - NCERT Maths Class 10

Go back to  'Ex.7.1'

Question

In a classroom, \(4\) friends are seated at the points \(A\)\(B\), \(C\) and \(D\) as shown in the following 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.

 

Text Solution

 

Reasoning:

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.

What is Known?

The \(x\) and \(y\) co-ordinates of the points between which the distance is to be measured can be deduced from the diagram.

What is Unknown?

To verify whether the positions of the four friends form a square or not.

Steps:

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,

\[\sqrt{\left(\mathrm{x}_{1}-\mathrm{x}_{2}\right)^{2}+\left(\mathrm{y}_{1}-\mathrm{y}_{2}\right)^{2}} \quad \ldots \text{Equation (1)}\]

To find \(AB\) i.e. Distance between Points \(A \;(3, 4)\) and \(B\; (6, 7)\)

  • \(x_1 = 3\)
  • \(y_1 = 4\)
  • \(x_2 = 6\)
  • \(y_2 =7\)

By substituting the values in the Equation (1), we get

\[\begin{align}AB &= \sqrt {{{(3 - 6)}^2} + {{(4 - 7)}^2}} \\ &= \sqrt {{{( - 3)}^2} + {{( - 3)}^2}} \\ &= \sqrt {9 + 9} \\ &= \sqrt {18} \\ &= 3\sqrt 2 \end{align}\]

To find \(BC\) i.e. Distance between Points \(B (6, 7)\) and \(C (9, 4)\)

  • \(x_1 = 6\)
  • \(y_1 = 7\)
  • \(x_2 = 9\)
  • \(y_2 = 4\)

By substituting the values in the Equation (1), we get

\[\begin{align}BC& = \sqrt {{{(6 - 9)}^2} + {{(7 - 4)}^2}} \\ &= \sqrt {{{( - 3)}^2} + {{(3)}^2}} \\ &= \sqrt {9 + 9} \\& = \sqrt {18} \\ &= 3\sqrt 2 \end{align}\]

To find \(CD\) i.e. Distance between Points \(C \; (9, 4)\) and \(D\; (6, 1)\)

  • \(x_1 = 9\)
  • \(y_1 = 4\)
  • \(x_2 = 6\)
  • \(y_2 = 1\)

By substituting the values in the Equation (1)

\[\begin{align}CB& = \sqrt {{{(9 - 6)}^2} + {{(4 - 1)}^2}} \\ &= \sqrt {{{(3)}^2} + {{(3)}^2}} \\ &= \sqrt {9 + 9} \\ &= \sqrt {18} \\ &= 3\sqrt 2 \end{align}\]

To find \(AD\) i.e. Distance between Points \(B\; (3, 4)\) and \(D \;(6, 1)\)

  • \(x_1 = 3\)
  • \(y_1 = 4\)
  • \(x_2 = 6\)
  • \(y_2 = 1\)

By substituting the values in the Equation (1)

\[\begin{align}AD &= \sqrt {{{(3 - 6)}^2} + {{(4 - 1)}^2}} \\ &= \sqrt {{{( - 3)}^2} + {{(3)}^2}} \\ &= \sqrt {9 + 9} \\&= \sqrt {18} \\ &= 3\sqrt 2 \end{align}\]

To find \(AC\) i.e. Distance between Points \(A \;(3, 4)\) and \(C\; (9, 4)\)

  • \(x_1 = 3\)
  • \(y_1 = 4\)
  • \(x_2 = 9\)
  • \(y_2 = 4\)

By substituting the values in the Equation (1), we get

\[\begin{align}{\text{Diagonal}} \; AC &= \sqrt {{{(3 - 9)}^2} + {{(4 - 4)}^2}} \\ &= \sqrt {{{( - 6)}^2} + {0^2}} \\ &= 6\end{align}\]

To find \(BD\) Distance between Points \(B \;(6, 7)\) and \(D \;(6, 1)\)

  • \(x_1 = 6\)
  • \(y_1 = 7\)
  • \(x_2 = 6\)
  • \(y_2 = 1\)

By substituting the values in the Equation (1)

\[\begin{align}\text{Diagonal} \;BD &= \sqrt {{{(6 - 6)}^2} + {{(7 - 1)}^2}} \\ &= \sqrt {{0^2} + {{( - 6)}^2}} \\ &= 6\end{align}\]

The four sides \(AB\), \(BC\), \(CD\), and \(AD\) are of same length and diagonals \(AC\) and \(BD\) are of equal length. Therefore, \(ABCD\) is a square and hence, Champa was correct

  

Frequently Asked Questions



What are Class 10 NCERT Exemplars?
While getting good scores in school tests is a desirable outcome, it is not a reliable indicator of how strong your child’s math foundation really is. Many students who score well in school exams in their earlier years, might struggle with math in higher grades because of a weak foundation. At Cuemath, we evaluate your child’s grasp of math fundamentals, and take corrective actions immediately. Also, your child may have limited exposure in their school, and in most cases, may not feel challenged to learn more. Cuemath's customised learning plan ensures your child is challenged with varied difficulty levels of questions at every stage.
What is the difference between CBSE and NCERT syllabus for Class 10?
How will Class 10 NCERT books help in exam preparation?
How will Class 10 NCERT books help you understand basic math concepts?
Which is the best video solution for the class 10 maths NCERT?