Coordinate Geometry：練習問題

A point \(S(-2, -5)\) is reflected in the \(y\)-axis to obtain its image \(S'\). Find the coordinates of \(S'\).

The point \(R(2, 5)\) is reflected in the line \(x = 3\). Find the coordinates of its image \(R'\).

Point \(P(a, b)\) is reflected across the y-axis to obtain \(P'\). Then \(P'\) is rotated \(90^\circ\) clockwise about the origin to obtain \(P''\). If the coordinates of \(P''\) are \((3, 4)\), what are the coordinates of the original point \(P\)?

Point \(A\) is located 4 units to the right of the y-axis and 7 units below the x-axis. What are the coordinates of point \(A\)?

Point \(M\) is the midpoint of the line segment joining \(A(-3, 8)\) and \(B(7, 2)\). Find the slope of the straight line passing through the origin \((0, 0)\) and point \(M\).

What is the x-coordinate of a point that lies on the y-axis?

Points A(3, -2) and B(9, 4) define a line segment. A point P divides AB internally in the ratio 2:1. If a straight line L passes through P and has a gradient of \(m = -\frac{1}{2}\), find the equation of line L in the form \(ax + by = c\).

Find the coordinates of the point of intersection of two lines, where line \(L_1\) has a slope of \(2\) and passes through \(A(1, 3)\), and line \(L_2\) is perpendicular to \(L_1\) and passes through \(B(4, 5)\).

Consider the points A\((-6, 4)\) and B\((2, 10)\) in a rectangular coordinate plane.

(a) Find the coordinates of the mid-point M of the line segment \(AB\).

(b) Calculate the slope of the line segment \(AB\).

(c) A straight line \(L\) is parallel to the line segment \(AB\) and passes through the point C\((0, -2)\). Find the \(y\)-intercept of the line \(L\).

Given three points $$A(2, 5)$$, $$B(6, 3)$$, and $$C(x, 7)$$.

If the angle $$\angle ABC$$ is a right angle ($$90^\circ$$), find the value of $$x$$.