Introduction to Coordinates：練習問題

If point \(A(k, -4)\) is at a distance of \(5\) units from the \(y\)-axis and lies in the third quadrant, what is the value of \(k\)?

In the rectangular coordinate plane, point \(M\) divides the line segment joining \(A(-5, 10)\) and \(B(5, 0)\) internally in the ratio \(1:4\). What are the coordinates of point \(M\)?

Point \(C(1, 0)\) lies on the straight line segment connecting \(A(-3, -2)\) and \(B(7, 3)\). In what ratio does \(C\) divide the line segment \(AB\)?

Point \(C(5, -2)\) is translated \(4\) units to the left and \(6\) units upwards. What are the coordinates of the resulting point \(C'\)?

A straight line has an \(x\)-intercept of \(4\) and a \(y\)-intercept of \(-2\). What is the slope of this line?

A point \(A\) is located on the \(y\)-axis and its distance from the origin is \(7\) units. If its \(y\)-coordinate is negative, what are the coordinates of point \(A\)?

A straight line passes through the points \(A(2, -3)\) and \(B(6, 3)\). Calculate the \(y\)-intercept of this line.

Let \( M \) be the midpoint of the line segment joining \( A(2, 4) \) and \( B(6, 8) \). If \( M \) is rotated \( 180^\circ \) about the origin to become \( M' \), find the coordinates of \( M' \) and calculate the distance \( OM' \).

Consider a point P with coordinates \((5, -7)\).

(a) In which quadrant does point P lie?
(b) Find the shortest distance from point P to the \(x\)-axis.
(c) If point Q is on the \(y\)-axis and has the same \(y\)-coordinate as point P, state the coordinates of Q.

Point \(P(3, -4)\) undergoes a series of transformations in the rectangular coordinate plane. First, it is reflected across the \(y\)-axis to become point \(P'\). Then, \(P'\) is rotated \(90^\circ\) clockwise about the origin \(O(0, 0)\) to become point \(P''\).

(a) State the coordinates of \(P'\) and \(P''\).
(b) Calculate the distance between the origin \(O\) and point \(P''\).
(c) Calculate the area of triangle \(OPP''\).