Question

16. Check whether the points P(0,4),(-2,0) and R (2,0) are the vertices of an equilateral
triangle.​

Answer 1

$\textbf{Given:}$

$\textsf{Points are P(0,4), Q(-2,0), R(2,0)}$

$\textbf{To check:}$

$\textsf{PQR is an equilateral triangle or not}$

$\textbf{Solution:}$

$\textsf{We find out lengths of the sides of the triangle PQR}$

$\textsf{by using distance formula}$

$\mathsf{PQ=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}}$

$\mathsf{PQ=\sqrt{(0+2)^2+(4-0)^2}}$

$\mathsf{PQ=\sqrt{4+16}}$

$\mathsf{PQ=\sqrt{20}}$

$\mathsf{QR=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}}$

$\mathsf{QR=\sqrt{(-2-2)^2+(0-0)^2}}$

$\mathsf{QR=\sqrt{16}}$

$\mathsf{QR=4}$

$\mathsf{PR=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}}$

$\mathsf{PR=\sqrt{(0-2)^2+(4-0)^2}}$

$\mathsf{PR=\sqrt{4+16}}$

$\mathsf{PR=\sqrt{20}}$

$\textsf{Clearly, all the sides are not equal}$

$\therefore\triangle\textsf{PQR is not an equilateral triangle}$

$\mathsf{But,\;PQ=PR}$

$\implies\triangle\textsf{PQR is an isoceles triangle}$