Question

The points (5, -2), (6, 4) and (7, -2) are the vertices of an isosceles triangle.
(A). True
(B). False

Answer 1

HINT: - The most important formula that would be used in solving this question is the distance formula that is given as follows
\[d=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}}\]
(Here \[\left( {{x}_{2}},{{y}_{2}} \right)\ and\ \left( {{x}_{1}},{{y}_{1}} \right)\] are the coordinates of the points between which the distance is being calculated)

Complete step-by-step solution -
In this question, we will first try to find the distance between the three points taking two at a time and then we will check whether the distance between points, that is whether any two sides are equal or not. If they are equal, then we can say that the coordinates are of the vertices of a triangle which is isosceles.
As mentioned in the question, we have to find whether the triangle with the given coordinates is an isosceles triangle or not.
Now, using the distance formula that is given in the hint, we can calculate the distance between any two points out of the three as follows
Let’s take (6, 4) and (7, -2) first as follows
 \[\begin{align}
  & {{d}_{1}}=\sqrt{{{\left( 6-7 \right)}^{2}}+{{(4-(-2))}^{2}}} \\
\Rightarrow & {{d}_{1}}=\sqrt{{{\left( -1 \right)}^{2}}+{{(4+2)}^{2}}} \\
\Rightarrow & {{d}_{1}}=\sqrt{1+{{(6)}^{2}}} \\
\Rightarrow & {{d}_{1}}=\sqrt{1+36} \\
 \Rightarrow & {{d}_{1}}=\sqrt{37} \\
\end{align}\]
Now, we will take (5, -2) and (6, 4) ass follows
\[\begin{align}
  & {{d}_{2}}=\sqrt{{{\left( 5-6 \right)}^{2}}+{{(-2-4)}^{2}}} \\
\Rightarrow & {{d}_{2}}=\sqrt{{{\left( -1 \right)}^{2}}+{{(-6)}^{2}}} \\
\Rightarrow & {{d}_{2}}=\sqrt{1+36} \\
\Rightarrow & {{d}_{2}}=\sqrt{37} \\
\end{align}\]
Now, for the third pair, we can write as follows for (5, -2) and (7, -2)
\[\begin{align}
  & {{d}_{3}}=\sqrt{{{\left( 5-7 \right)}^{2}}+{{(-2-(-2))}^{2}}} \\
\Rightarrow & {{d}_{3}}=\sqrt{{{\left( -2 \right)}^{2}}} \\
\Rightarrow & {{d}_{3}}=\sqrt{4}=2 \\
\end{align}\]
Now, as the length of only two sides is equal, hence, the triangle that is formed from these points as the vertices is an isosceles triangle and not an equilateral triangle.

NOTE:- The students can make an error if they don’t know the distance formula that is given in the hint. Also, if the triangle would have been an equilateral triangle and if the student would have stopped by finding only two sides equal without calculating the length of the third side, then the answer would have been wrong.