Coordinate geometry：練習問題

A line segment passes through the points \(A(2, 3)\) and \(B(5, 9)\). Calculate the gradient of the line segment \(AB\).

Find the equation of the straight line that passes through the points \( (2, 3) \) and \( (4, 7) \) in the form \( y = mx + c \).

The points \(P\) and \(Q\) have coordinates \((0, 4)\) and \((6, 0)\) respectively. Find the equation of the perpendicular bisector of the line segment \(PQ\).

Find the equation of the straight line that passes through the point \((1, 2)\) and has a gradient of \(-3\).

The point \( P(k, 4) \) lies on the line perpendicular to the line \( 2x + 3y = 6 \) and passing through the point \( (2, 1) \). Find the value of \( k \).

Find the gradient of the straight line that passes through the points \(A(2, 5)\) and \(B(-1, 14)\).

The line \( L_1 \) passes through the points \( A(2, -3) \) and \( B(6, 5) \). Find the equation of the line \( L_2 \) which is the perpendicular bisector of the segment \( AB \).

The points \( A \) and \( B \) have coordinates \( (-3, 2) \) and \( (5, 8) \) respectively.
(a) Find the length of the line segment \( AB \).
(b) Find the equation of the line \( L_1 \) which passes through \( A \) and \( B \).
(c) The line \( L_2 \) is perpendicular to \( L_1 \) and passes through the point \( (2, -1) \). Find the equation of \( L_2 \).
(d) Find the coordinates of the point where \( L_2 \) crosses the \( y \)-axis.

The coordinates of three points are \( A(1, 2) \), \( B(5, 8) \) and \( C(9, 2) \).
(a) Find the equation of the line \( L \) that passes through \( B \) and is parallel to \( AC \).
(b) Find the equation of the perpendicular bisector of the line segment \( AB \).
(c) Find the coordinates of the point \( D \) where the perpendicular bisector of \( AB \) intersects the line \( L \).
(d) Calculate the area of the triangle \( ABC \).