Calculate the distance from the point (3,5) to the line
y = x + 4.

Question

Calculate the distance from the point (3,5) to the line 
y = x + 4.

anonymous · Answer

choose a point on y = x+ 4 (0,4)

now use distance formula for two points of (0,4) and (3,5)

anonymous · Answer

|dw:1332106712053:dw| The shortest distance will be along a line that is perpendicular to the given line. That line will have the opposite, reciprocal slope, so it would be y = -x +b. To find its b value, plug in the point's coordinates:$$5 = -3+b$$$$b =8$$ So the perpendicular line that passes through the point is y = -x+8. To find the location on the original line that is closest to the given point, find the intersection between the two lines:$$x+4 = -x+8$$$$2x = 4$$$$x =2$$ Then plug in 2 for x in either equation to find the y value:$$y = x+ 4 = 2+4 = 6$$ So the closest point on the original line to the given point is (2, 6). Now use the distance formula to find the distance between the two points$$d = \sqrt{(\Delta x)^{2} + (\Delta y)^{2}} = \sqrt{(3-2)^{2} + (5-6)^{2}} = \sqrt{2}$$