Ben
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Find the distance from the point [tex](6, 4)[/tex] to the line [tex]y=x+4[/tex].

Respuesta :

znk

Answer:

[tex]\large \boxed{3\sqrt{2}}[/tex]

Step-by-step explanation:

1. Express the line in standard form.

ax + by + c = 0

[tex]\begin{array}{rcl}y & = & x + 4\\-x + y - 4 & = & 0\\\end{array}[/tex]

2. Calculate the distance

The formula for the distance d from a point (x, y) and the line is:

[tex]d = \dfrac{|ax + by + c|}{\sqrt{a^{2} + b^{2}}}[/tex]

Insert the values: a = -1; b = 1; c = -4; x = 6; y = 4

[tex]\begin{array}{rcl}d &= &\dfrac{|(-1)(6) + 1\times4 + (-4)|}{\sqrt{(-1)^{2} + 1^{2}}}\\\\& = & \dfrac{|-6 + 4 - 4|}{\sqrt{1 + 1}}\\\\& = & \dfrac{|-6|}{\sqrt{2}}\\\\& = &\dfrac{6}{\sqrt{2}}\\\\& = & \dfrac{3\times\sqrt{2}\times\sqrt{2}}{\sqrt{2}}\\\\& = & \mathbf{3{\sqrt{2}}}\end{array}\\\text{The distance from the point to the line is $\large \boxed{\mathbf{3\sqrt{2}}}$}[/tex]

Wolfyy

Hey!

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First, we need to find the second coordinate.

[tex]\text{We know that the slope is 1 and the y-intercept is 4.}\\\text{We use}~\frac{1 - 4}{x - 6} = 1~\text{to solve for x.}\\\text{We can simplify this to}~\frac{-3}{x-6}\\\text{Now we know that whatever x subtracts to 6 we get -3 to get a slope of 1.}\\\text{After solving we can use 3 for x.}\\\text{We subsitute for x and end of getting}~\frac{-3}{-3}~\text{which equals 1.}\\\text{The second point is (3, 4)}[/tex]

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Second, we need to find the distance between both points.

[tex]d = \sqrt{(3 - 6)^2 + (1 - 4)^2} \\d = \sqrt{(-3)^2 + (-3)^2} \\d = \sqrt{9+9} \\d = \sqrt{18} \\\text{We can simplify this using what we know about radicals.}\\d = 3\sqrt{2}[/tex]

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Hence, the answer is [tex]3\sqrt{2}[/tex]!

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Hope This Helped! Good Luck!

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