# 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.

Video Solution
Coordinate Geometry
Ex 7.1 | Question 4

## 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( {{x_{\text{1}}} - {x_{\text{2}}}} \right)}^2} + {{\left( {{y_{\text{1}}} - {y_{\text{2}}}} \right)}^2}} \;\;...(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.

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