Verified answer

The shortest distance between a point and a line is a perpendicular line, which you have already found. There are two ways to do this.

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1). Use the distance formula given by:

d = |y1 - m*x1 - b| / √(m² + 1), where

(x1, y1) = (6,-2), and m and b are the slope and y-intercept of the line.

−2x+4y=0

4y = 2x

y = ½x ---------> m = ½, b = 0

d = |y1 - m*x1 - b| / √(m² + 1)

= |-2 - 3 - 0| / √1.25

= 5/√(5/4)

= 5*√(4/5)

= 5*2/√5

d = 2√5

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2) Find the length of the segment of the perpendicular line y= -2x + 10 from (6, -2) to the intersection of the two lines.

Two lines intersect at a point where they have the same coordinates, so set the two y equations equal to each other and solve for x. Then plug that x into one of the equations to find the y coordinate.

−2x+4y=0 ------------> y = ½x

½x = -2x + 10

2½ x = 10

5/2 x = 10

x = 4

-------> y = ½x = 2

-------> (4, 2) is the intersection point.

So find the distance between the points (6, -2) and (4, 2).

Distance formula:

d = √[(x2 - x1)² + (y2 - y1)²]

= √[(4-6)² + (2 - (-2))²]

= √(4 + 16)

= √20

= 2√5