Question

Line t contains points (1,3) and (2,2). point r has coordinates (4,4). find the distance from t to r.

Answer 1

`\begin{gathered} \text{ First, we construct the equation of the line t, } \\ \text{ the slope of the line is } \\ m\text{ =}\frac{y_{2\text{ }}-y_1}{x_2-x_1}\text{,} \end{gathered}`
`\begin{gathered} \text{Then} \\ m\text{ = -1} \\ so\text{ y = -x + b, replacing x and y, at any point of the line t, we find b} \end{gathered}`
`\begin{gathered} 3=-(1)+b \\ 4=b \\ \text{ Then, the equation of the line t is } \\ y=-x+4 \end{gathered}`
`\begin{gathered} \text{ The equation of the distance of a point }A=(x,y)\text{ to a line t:y = ax + b is} \\ d(A,t\text{)}=\frac{\sqrt[]{a^2+1}} \\ d(A,t)=\frac(-1\cdot4)-(4)+4{\sqrt[]{1+1}} \\ d(A,t)=\frac{4}{\sqrt[]{2}} \end{gathered}`
`\begin{gathered} \text{Simplifying this expression, we get} \\ \\ d(A,t)=2\sqrt[]{2} \end{gathered}`