Locus of Mid Point: X-Axis Cut Points

[phymatter]

Homework Statement

From the point (6,-8) all possible lines are drawn to cut x-axis , find the locus of their middle ponts

Homework Equations

none

The Attempt at a Solution

i got the coordinates of the middle point as ((6+x)/2 ,-4 ) , but what will the locus be ?

 

[Mark44]

The expression you have gives you the locus. What do all of the midpoints have in common?

 

[phymatter]

The expression you have gives you the locus. What do all of the midpoints have in common?

you mean that the locus is y=-4 neglecting the term containing x , but why are we neglecting x ? i mean if we put different values of x we get different mid points , so why are we neglecting it??
pls. help , its getting over my head :((

 

[verty]

Draw a right angled triangle and join the midpoints of two sides. This new line will be parallel to the third side. Similarly, in any triangle, the line between the midpoints of two sides is parallel to the third side. You can prove this because the smaller triangle is similar to the larger, so the corresponding angles are equal.

If the apex is (6,-8) and the third side is the x-axis, the line between the midpoints is the locus. Does this convince you?

 

[VietDao29]

you mean that the locus is y=-4 neglecting the term containing x , but why are we neglecting x ? i mean if we put different values of x we get different mid points , so why are we neglecting it??
pls. help , its getting over my head :((

You're neglecting the x, because the image of $$\frac{6 + x}{2}$$ spreads over the real number. What I mean is:

$$\forall x_0 \in \mathbb{R} , \exists x \in \mathbb{R} : \frac{6 + x}{2} = x_0$$.

So, take any real number x₀, the point (x₀; -4) is the midpoint of the line segment, of which 2 ends are (6; -8), and $$\left( 2x_0 - 6; 0 \right)$$.

Hope I'm being clear enough. :)

 

[HallsofIvy]

All points of the form ((6+x)/2 ,-4 ) satisfy y= -4 and that is the equation of the locus, a horizontal line. The "(6+x)/2" gives the x coordinate of a point on that line (for x being the x coordinate of the point (x, 0) that the line from (6, -8) is drawn through).