Question

Find the slope of the line which passes through the origin, and the mid-point of the line segment joining the points A (0, -4) and B (8, 0).

Answer 1

Hint: We know that the slope of a line joining the two points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] is given by as follows:
\[slope=\tan \theta =\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]
We will also use the mid-point formula for the two points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] given by as follows:
x-coordinate \[=\dfrac{{{x}_{1}}+{{x}_{2}}}{2}\]
y-coordinate \[=\dfrac{{{y}_{1}}+{{y}_{2}}}{2}\]

Complete step-by-step answer:
We have been asked to find the slope of a line which passes through the origin and the mid-point of the line segment joining the points A (0, -4) and B (8, 0).
We know that the mid-point formula for the two points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] given by as follows:
x-coordinate \[=\dfrac{{{x}_{1}}+{{x}_{2}}}{2}\]
y-coordinate \[=\dfrac{{{y}_{1}}+{{y}_{2}}}{2}\]
We have \[{{x}_{1}}=0,{{x}_{2}}=8,{{y}_{1}}=-4,{{y}_{2}}=0\]
By using the mid-point formula, we have as follows:
x-coordinate \[=\dfrac{0+8}{2}=4\]
y-coordinate \[=\dfrac{-4+0}{2}=-2\]
Thus the mid-point is (4, -2).
Now we have been given that the line is passing through the origin and mid-point of the line segment joining the points A (0, -4) and B (8, 0), i.e. the line passes through (0, 0) and (4, -2).
We know that if the line passes through \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] then its slope is given by as follows:
\[slope=\tan \theta =\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]
So we have \[{{x}_{1}}=0,{{x}_{2}}=4,{{y}_{1}}=0,{{y}_{2}}=-2\]
\[\Rightarrow slope=\dfrac{-2-0}{4-0}=\dfrac{-2}{4}=\dfrac{-1}{2}\]
Therefore, the required slope of the line is equal to \[\left( \dfrac{-1}{2} \right)\].

Note: Just be careful while substituting the values of \[{{x}_{1}},{{x}_{2}},{{y}_{1}},{{y}_{2}}\] in the formula to find the slope because if you substitute it incorrectly you will get the wrong answer. Also remember that the origin is a point whose x-coordinate as well as y-coordinate are equal to zero. If a line passes through origin and a point \[\left( {{x}_{1}},{{y}_{1}} \right)\] then its slope is equal to \[\left( \dfrac{{{y}_{1}}}{{{x}_{1}}} \right)\].