Find the distance from the point (−10,−5) to the line y = −5x − 3.

Question

Find the distance from the point (−10,−5) to the line y = −5x − 3.

Answer 1

There is a formula for finding the distance from a point to a line.

The following graph shows how to use that formula AND how to find the distance by finding equation of a line perpendicular to the given line, passing through the given point, finding the point of intersection, and finally find the distance from the point of intersection to the given point.

Using the formula is a lot quicker, but perhaps not as challenging?!

If ax+by+c = 0 is the equation of the line, and the given point is (x₁,y_1), then

D=| (ax₁+by₁+c) | / √(a²+b²) .

In this problem a=5, b=1,c=3, x₁=-10, y₁=-5.

So, |5(-10) + 1(-5) + 3| = |-52| = 52

√(25+1) = √26

D= 52/√26

desmos.com/calculator/vrmxjssa3z

Answer 2

The shortest distance from the point to the line is a perpendicular line. We need to find the distance from from the point to where the perpendicular line intersects the other line.

Find the equation of that perpendicular line. We have a point on that line (-10,5). The slope that line will be the negative reciprocal of the slope of the line y = -5x -3, which would be 1/5.

so the equation of the perpendicular line is:

y - 5= 1/5 (x+10)

y = 1/5x + 2 + 5

y = 1/5x + 7

Now equate the equations of the original line and perpendicular line to find their intersection:

1/5x + 7 = -5x - 3

x + 35 = -25x - 15

26x = -50

x = -1.92

Plug in the value of x into either equation to get the value of y:

y = -5x - 3, so y = -5(-1.92) - 3 = 6.6

So point of intersection is (-1.92,6.6)

Now find distance between (-1.92,6.6) and (-10,5)

sqrt((-10-(-1.92))^2 + (5-6.6)^) = 8.2