Question

question 4
in the diagram below, the circle with mid - point m is drawn.
the equation of line jk is given as y = x - 8.
point k is the y - intercept of line jk. line pm is parallel to the x - axis.
qm = x + 2 units. pq is 3 times smaller than qm.
s and j(8;0) are points on the x - axis. r is a point on the circumference of the circle.
o(0;0)
s
j(8;0)
p(0; - 4)
q
x + 2
m
r
k
4.1 determine the coordinates of the mid - point, m.

Explanation:

Step1: Find the y - coordinate of M

Since line PM is parallel to the x - axis and P(0, - 4), the y - coordinate of M is the same as the y - coordinate of P. So, $y_M=-4$.

Step2: Find the x - coordinate of M

We know that PQ is 3 times smaller than QM. Let $QM = x + 2$, then $PQ=\frac{x + 2}{3}$. Also, the x - coordinate of P is 0. The x - coordinate of M, $x_M$ can be found from the relationship on the x - axis. Since P(0,-4) and M has x - coordinate based on the length from P along the horizontal line (parallel to x - axis). We know that the distance from P to M along the x - axis is QM. The x - coordinate of M, $x_M$:
We know that the line JK is $y=x - 8$ and K is the y - intercept, so K(0,-8).
Since PM is parallel to the x - axis and P(0,-4), and we know that the circle is centered at M.
We note that from the figure, if we consider the horizontal distance from P to M. Let's assume the x - coordinate of M is $x$. We know that $QM=x + 2$.
Since P(0,-4) and M has x - coordinate $x$ and y - coordinate - 4.
We know that the distance from P to M along the x - axis gives us the x - coordinate of M.
Since $PQ+QM$ gives the x - coordinate of M (starting from P(0,-4)). And $PQ=\frac{QM}{3}$, let $QM = d=x + 2$. Then $PQ=\frac{d}{3}$.
We know that the x - coordinate of M:
Since P(0,-4) and M is on the line $y=-4$, and from the given lengths, if we consider the horizontal displacement from P to M.
We know that $QM=x + 2$. And since P(0,-4), the x - coordinate of M is $x_M = 4$.
So the coordinates of M are (4,-4).

Answer:

(4,-4)