Answer:
Calculate the distance between two points on a number line when both are non-negative.
Calculate the distance between two points on a number line when at least one is negative.
Step-by-step explanation:
The minimum distance from the point to the line would be found by drawing a segment perpendicular to the line directly to the point. Our first step is to find the equation of the new line that connects the point to the line given in the problem. Because we know this new line is perpendicular to the line we're finding the distance to, we know its slope will be the negative inverse of the line its perpendicular to. So if the line we're finding the distance to is:
f(x)=−13x+2
Then its slope is -1/3, so the slope of a line perpendicular to it would be 3. Now that we know the slope of the line that will give the shortest distance from the point to the given line, we can plug the coordinates of our point into the formula for a line to get the full equation of the new line:
f(x)=mx+b
f(x)=3x+b
−4=3(−2)+b→b=2
f(x)=3x+2
Now that we know the equation of our perpendicular line, our next step is to find the point where it intersects the line given in the problem:
3x+2=−13x+2
103x=0→x=0
So if the lines intersect at x=0, we plug that value into either equation to find the y coordinate of the point where the lines intersect, which is the point on the line closest to the point given in the problem and therefore tells us the location of the minimum distance from the point to the line:
f(0)=3(0)+2=2
So we now know we want to find the distance between the following two points:
(−2,−4) and (0,2)
Using the following formula for the distance between two points, which we can see is just an application of the Pythagorean Theorem, we can plug in the values of our two points and calculate the shortest distance between the point and line given in the problem:
d=(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√
d=(0−(−2))2+(2−(−4))2−−−−−−−−−−−−−−−−−−−−√=(2)2+(6)2−−−−−−−−−√=40−−√
Which we can then simplify by factoring the radical:
40−−√=4⋅10−−−−√=210−−√
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