Find the distance from the point (5, -2) to the line 3x - 4y = 12 by using the s

Question

Find the distance from the point (5, -2) to the line 3x - 4y = 12 by using the steps listed below: Write the equation of the line perpendicular to 3x - 4y = 12 and passing through the point (5, -2). Write the line in the form: ax + by = c where a > 0 and a, b, & c are integers. Show work. Line: _____ Find the point of intersection of the line found in step 1 and 3x - 4y = 12. Show work below. Point: _____ Find the distance from (5, -2) to the point found in step 2. Show work below. Distance: _____

Explanation / Answer

1) 3x - 4y = 12

Or, 4y = 3x - 12

Or, y = 3/4x - 3

So the slope of the given line is 3/4

Slope of the line which is perpendicular to the given line 3x - 4y =12 is -4/3

Now the equation of the line passing through the point (5, -2) is y - (-2) = -4/3(x - 5)

Or, y + 2 = -4/3x + 20/3

Or, y = -4/3x + 14/3

2) from the two equations we will get,

3/4x - 3 = -4/3x + 14/3

Or, 3/4x + 4/3x = 14/3 + 3

Or, (25/12)x = 23/3

Or, x = (23/3) * (12/25)

Or, x = 92/25

y = (3/4) * (92/25) - 3

Or, y = 69/25 - 3

Or, y = -6/25

So, the point of intersection is (92/25, -6/25)

3) The distance between the points (5, -2) and (92/25, -6/25) = sqrt[ (92/25 - 5)2 + (-6/25 - (-2))2 ]

= Sqrt(1.7424 + 3.0976)

= Sqrt(4.84) = 2.2

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