Question

what is the shortest distance between the line ( y = 2x + 4 ) and the point at ( (0, 0) )? round to the nearest tenth.

1.8
2.0
3.2
4.0

Explanation:

Step1: Convert the line equation to standard form

The line equation \(y = 2x+4\) can be rewritten as \(2x - y+4 = 0\). The formula for the distance \(d\) from a point \((x_0,y_0)\) to a line \(Ax + By+C = 0\) is \(d=\frac{\vert Ax_0+By_0 + C\vert}{\sqrt{A^{2}+B^{2}}}\). Here, \(A = 2\), \(B=- 1\), \(C = 4\), \(x_0=0\), \(y_0 = 0\).

Step2: Substitute values into the distance formula

Substitute the values into the formula: \(d=\frac{\vert2\times0+( - 1)\times0 + 4\vert}{\sqrt{2^{2}+(-1)^{2}}}=\frac{\vert4\vert}{\sqrt{4 + 1}}=\frac{4}{\sqrt{5}}\).

Step3: Simplify the expression

\(\frac{4}{\sqrt{5}}=\frac{4\sqrt{5}}{5}\approx\frac{4\times2.236}{5}=\frac{8.944}{5}=1.7888\approx1.8\)

Answer:

1.8