Answer:
The distance between the point P(0, 0) and the line y = −x + 4 is [tex]2\sqrt{2}[/tex] units.
Step-by-step explanation:
Distance between the point [tex](x_0,y_0)[/tex] and the line [tex]ax+by+c=0[/tex] is
[tex]d=\dfrac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}[/tex]
The given equation is
[tex]y=-x+4[/tex]
Taking all terms on the left side.
[tex]x+y-4=0[/tex]
The distance between the point [tex](0,0)[/tex] and the line [tex]x+y-4=0[/tex] is
[tex]d=\dfrac{|0+0-4|}{\sqrt{1^2+1^2}}[/tex]
[tex]d=\dfrac{|-4|}{\sqrt{2}}[/tex]
[tex]d=\dfrac{4}{\sqrt{2}}[/tex]
Rationalize denominator.
[tex]d=\dfrac{4}{\sqrt{2}}\times \dfrac{\sqrt{2}}{\sqrt{2}}[/tex]
[tex]d=\dfrac{4\sqrt{2}}{2}[/tex]
[tex]d=2\sqrt{2}[/tex]
Therefore, the distance between the point P(0, 0) and the line y = −x + 4 is [tex]2\sqrt{2}[/tex] units.
