The Cartesian Plane and Coordinate Geometry：练习题

Find the coordinates of the midpoint of the line segment joining the points \(A(2, -5)\) and \(B(6, 11)\).

The distance between the point \(M(k, 2)\) and the point \(N(3, -4)\) is \(10\) units. Find the possible values of \(k\).

In the coordinate plane, the vertices of a triangle are \(A(1, 2)\), \(B(5, 2)\), and \(C(3, 6)\). If the triangle is reflected across the line \(y = x\) to form triangle \(A'B'C'\), find the coordinates of the centroid of \(A'B'C'\).

A straight line passes through the points \(P(-2, 3)\) and \(Q(4, -1)\). Find the equation of the line in the form \(ax + by + c = 0\), where \(a\) is a positive integer.

A line \(L_1\) passes through the points \(R(-1, 4)\) and \(S(2, 10)\). Another line \(L_2\) is perpendicular to \(L_1\) and passes through the midpoint of \(RS\). Find the \(y\)-intercept of \(L_2\).

Find the coordinates of the mid-point of the line segment joining \(P(-4, 7)\) and \(Q(2, -1)\).

Show that the triangle with vertices \(A(0, 0)\), \(B(4, 0)\), and \(C(0, 3)\) is a right-angled triangle by calculating the slopes of its sides.

Point \(P\) divides the line segment joining \(A(-2, 5)\) and \(B(3, -5)\) internally in the ratio \(2:3\). Find the coordinates of \(P\).

A line segment connects points \(A(2, 5)\) and \(B(8, 13)\).
(a) Calculate the coordinates of the midpoint \(M\) of the line segment \(AB\).
(b) Determine the length of the line segment \(AB\).

A straight line \(L_1\) passes through the points \(C(0, -4)\) and \(D(6, 0)\).
(a) Find the slope of line \(L_1\).
(b) Another line \(L_2\) is perpendicular to \(L_1\) and passes through the midpoint of \(CD\). Find the equation of line \(L_2\) in the form \(y = mx + c\).
(c) Find the \(x\)-intercept of line \(L_2\).