Question

The points \[A\left( {1,3} \right)\] and \[C\left( {5,1} \right)\] are the opposite vertices of rectangle. Then find the equation of the line passing through the other two vertices and of gradient 2.
A. \[2x + y - 8 = 0\]
B. \[2x - y - 4 = 0\]
C. \[2x - y + 4 = 0\]
D. \[2x + y + 7 = 0\]

Answer 1

Hint In the given question, two vertices of a rectangle are given. By using the mid-point formula, we will find the point of intersection of the diagonals of a rectangle. Then by substituting midpoint values and gradient in the point-slope form of a linear equation, we will find the equation of the line passing through the other two vertices.

Formula used
Midpoint Formula: The midpoint between the two points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] is: \[m = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)\]
Point-slope form of a linear equation of a line that passes through the point \[\left( {{x_1},{y_1}} \right)\] with slope \[m\] is: \[y - {y_1} = m\left( {x - {x_1}} \right)\]

Complete step by step solution:
The given opposite vertices of a rectangle are \[A\left( {1,3} \right)\] and \[C\left( {5,1} \right)\].
We know that the diagonals of a rectangle bisect each other.
Apply mid-point formula to calculate the mid-point of diagonals of the passing through the points \[A\left( {1,3} \right)\] and \[C\left( {5,1} \right)\].
Let \[M\left( {{x_1},{y_1}} \right)\] be the midpoint.
\[\left( {{x_1},{y_1}} \right) = \left( {\dfrac{{1 + 5}}{2},\dfrac{{3 + 1}}{2}} \right)\]
\[ \Rightarrow \]\[\left( {{x_1},{y_1}} \right) = \left( {\dfrac{6}{2},\dfrac{4}{2}} \right)\]
\[ \Rightarrow \]\[\left( {{x_1},{y_1}} \right) = \left( {3,2} \right)\]

The mid-point also lies on another diagonal with gradient 2.
Apply point-slope form of a linear equation of a line.
\[y - 2 = 2\left( {x - 3} \right)\]
Simplify the above equation.
\[y - 2 = 2x - 6\]
\[ \Rightarrow \]\[2x - y - 4 = 0\]

Hence the correct option is option B.

Note: If we know the slope of an equation, then we can let the equation in the form \[y = mx + c\] where \[m\] is slope or gradient of the line. Here \[c\] is only unknown. If we know a point on the line, then we can easily find the equation of the line. In the given question, we can let the required as \[y = 2x + c\]. Then put \[x = 3\] and \[y = 2\] in the equation to get the value of \[c\]. This is another way to solve the question.