Find a line that is perpendicular from x=10 and one point, (4,5)

Question

The title is quite self-explainary ;)

I know that the answer is $y=5$, but I am not sure how does one person come to that conclusions showing all the steps. Can you please help me here?

Answer 1

You could do this in many ways.

1) It is obvious for some.

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2) You can visualize this on your head, or even plot this graph and see clearly.

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3) This is the only method that I could think at this time of which you could present if you were asked to prove it.

$x= 10$ is a vertical line, A perpendicular line to this must be parallel to the x axis. Hence it must have a gradient 0. So it is in the form $y= (0)x+ c=c$

We know two points on this line we are finding which are :

$(4,5)$ and $(10,b)$

Where b is an unknown

So let's apply the formula for the gradient :

$$\frac{5-b}{4-10}=0$$

$$b=5$$

Now we know the line is in the form $$y=c$$ and since $y=5$ (of any of the two points) we can conclude that the line is $$y=5$$

Answer 2

I had to do my own version, using @Tharindu's knowledge for my assignment. Here it is if anyone wants a more detailed version. The coordinates are different, since I had a different question:

We need to find an equation that will create a line that is perpendicular to the equation $x=3.58$, given that one point on the line B. As shown in steps beforehand, the point $B$ is equal to $(14.99,8.28)$

Any equation in the form $x=c$ will always make a vertical line. A perpendicular line is a line that intersects on a right angle. Therefore, knowing that $x=3.58$ will always be a vertical line, the perpendicular line will a horizontal line. A horizontal line always has a $m$ value of 0.

Therefore, using the base equation for a linear equation: $$y=mx+c$$ Sub in the known values : $$y=(0)x+c$$ Simplify $$y=c$$

That is the equation of our horizontal line as of now. Now, we know one point on this horizontal line, $B$, and we know the $m$ value of the equation is 0.

With this information, we can find the equation of the line using the point-slope formula: $$y−y_1=m(x−x_1)$$

Where $y_1$ and $x_1$ can be any point on the line. In this case, we will use the $B$ coordinate.
Co-ordinate of B: $(14.99,8.28)$
Let x=14.99 be $x_1$
Let y=8.28 be $y_1$
$(14.99,8.28)$=$(x_1,y_1)$

Point-Slope Formula: $$y−y1=m(x−x1)$$

Sub in known values: $$y−y1=m(x−x1)$$ $$y−8.28=m(x−14.99)$$ $$y−8.28=0$$ $$y=8.28$$ Therefore, the equation that is perpendicular to the directrix, given that one point on that line has the coordinate (14.99,8.28), is y=8.28