Ex.7.1 Q4 Coordinate Geometry Solution - NCERT Maths Class 10

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Question

Check whether \((5, -2)\), \((6, 4)\) and \((7, -2)\) are the vertices of an isosceles triangle.

Text Solution

 

Reasoning:

An isosceles triangle is a triangle that has two sides of equal length.

To check whether the given points are vertices of an isosceles triangle, the distance between any of the \(2\) points should be the same for two pairs of points.

What is Known?

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

What is Unknown?

To check whether the given points are the vertices of an isosceles triangle.

Steps:

Let the points \((5, -2)\), \((6, 4)\), and \((7, -2)\) represent the vertices \(A\), \(B\), and \(C\) of the given triangle respectively.

We know that the distance between the two points is given by the Distance Formula,

Distance Formula \(\begin{align} = \sqrt {{{\left( {{{\text{x}}_{\text{1}}} - {{\text{x}}_{\text{2}}}} \right)}^2} + {{\left( {{{\text{y}}_{\text{1}}} - {{\text{y}}_{\text{2}}}} \right)}^2}} \qquad \qquad ...\,\;(1)\end{align}\)

To find \(AB\) i.e. Distance between Points \(A\; (5, -2)\) and \(B \;(6, 4)\)

  • \(x_1 = 5\)
  • \(y_1 = 2\)
  • \(x_2 = 6\)
  • \(y_2 = 4\)

By substituting the values in the Equation (1)

\[\begin{align}AB &= \sqrt {{{(5 - 6)}^2} + {{( - 2 - 4)}^2}} \\ &= \sqrt {{{( - 1)}^2} + {{( - 6)}^2}} \\ &= \sqrt {1 + 36} \\ &= \sqrt {37} \end{align}\]

To find \(BC\) Distance between Points \(B \;(6, 4)\) and \(C \;(7, -2)\)

  • \(x_1 = 6\)
  • \(y_1 = 4\)
  • \(x_2 = 7\)
  • \(y_2 = -2\)

By substituting the values in the Equation (1)

\[\begin{align}BC &= \sqrt {{{(6 - 7)}^2} + {{(4 - ( - 2))}^2}} \\ &= \sqrt {{{( - 1)}^2} + {{(6)}^2}} \\ &= \sqrt {1 + 36} \\ &= \sqrt {37} \end{align}\]

To find \(AC\) i.e. Distance between Points \(A \;(5, -2)\) and \(C \;(7, -2)\)

  • \(x_1 = 5\)
  • \(y_1 = -2\)
  • \(x_2 = 7\)
  • \(y_2 = -2\)

\[\begin{align}CA& = \sqrt {{{(5 - 7)}^2} + {{( - 2 - ( - 2))}^2}} \\ &= \sqrt {{{( - 2)}^2} + {0^2}} \\ &= 2\end{align}\]

From the above values of \(AB\), \(BC\) and \(AC\) we can conclude that \(AB\) \(=\) \(BC\). As, two sides are equal in length, therefore, \(ABC\) is an isosceles triangle.