\displaystyle \textbf{1. Distance Formula}
\displaystyle \text{The distance between two points }P(x_1,y_1)\text{ and }Q(x_2,y_2)\text{ is given by}
\displaystyle PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
\displaystyle \text{i.e., }
\displaystyle PQ=\sqrt{(\text{Difference of abscissae})^2+(\text{Difference of ordinates})^2}
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\displaystyle \textbf{2. Distance from the Origin}
\displaystyle \text{The distance of a point }P(x,y)\text{ from the origin }O(0,0)\text{ is given by}
\displaystyle OP=\sqrt{x^2+y^2}
\\

\displaystyle \textbf{3. Area of a Triangle}
\displaystyle \text{The area of the triangle whose vertices are }(x_1,y_1),\,(x_2,y_2)\text{ and }(x_3,y_3)
\displaystyle \text{is the absolute value of}
\displaystyle \frac{1}{2}\left[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right]
\displaystyle \text{or}
\displaystyle \frac{1}{2}\left|\begin{array}{ccc}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{array}\right|
\\

\displaystyle \textbf{4. Condition for Collinearity}
\displaystyle \text{If the points }(x_1,y_1),\,(x_2,y_2)\text{ and }(x_3,y_3)\text{ are collinear, then}
\displaystyle \left|\begin{array}{ccc}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{array}\right|=0
\\

\displaystyle \textbf{5. Section Formula}
\displaystyle \text{The coordinates of the point dividing the line segment joining }(x_1,y_1)\text{ and }(x_2,y_2)
\displaystyle \text{in the ratio }m:n\text{ are}
\displaystyle \left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)\text{ internally,}
\displaystyle \left(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n}\right)\text{ externally.}
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\displaystyle \textbf{6. Mid-point Formula}
\displaystyle \text{The coordinates of the mid-point of the line segment joining }(x_1,y_1)\text{ and }(x_2,y_2)\text{ are}
\displaystyle \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)
\\

\displaystyle \textbf{7. Centroid of a Triangle}
\displaystyle \text{The coordinates of the centroid of the triangle whose vertices are }(x_1,y_1),\,(x_2,y_2)\text{ and }(x_3,y_3)
\displaystyle \text{are}
\displaystyle \left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)
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