Question

triangles abd and ace are similar right triangles. which best describes how to find the equation of the line?
(0,3) a 2 d e
c(x,y)
b
4
\\(\frac{da}{bd} = \frac{ce}{ea}\\) so \\(\frac{2}{4} = \frac{y - 3}{x}\\); solve for y to get \\(y = \frac{1}{2}x + 3\\).
\\(\frac{da}{bd} = \frac{ea}{ce}\\) so \\(\frac{2}{4} = \frac{x - 2}{y}\\); solve for y to get \\(y = 2x - 4\\).
\\(\frac{bd}{da} = \frac{ce}{ea}\\) so \\(\frac{4}{2} = \frac{y - 3}{x - 2}\\); solve for y to get \\(y = 2x - 1\\).
\\(\frac{bd}{da} = \frac{ce}{ea}\\) so \\(\frac{4}{2} = \frac{y - 3}{x}\\); solve for y to get \\(y = 2x + 3\\).

Explanation:

Step1: Analyze similar triangles' ratios

For similar right triangles \(ABD\) and \(ACE\), the ratios of corresponding sides are equal. \(BD = 4\), \(DA = 2\), \(CE=y - 3\), \(EA=x\) (since \(A\) is at \((0,3)\) and \(C\) is at \((x,y)\), so the horizontal distance from \(A\) to \(C\) is \(x\) and vertical distance is \(y - 3\)). The ratio of \(BD\) to \(DA\) should equal the ratio of \(CE\) to \(EA\), so \(\frac{BD}{DA}=\frac{CE}{EA}\).

Step2: Substitute values into the ratio

Substitute \(BD = 4\), \(DA = 2\), \(CE=y - 3\), \(EA=x\) into \(\frac{BD}{DA}=\frac{CE}{EA}\), we get \(\frac{4}{2}=\frac{y - 3}{x}\).

Step3: Solve the equation for \(y\)

Simplify \(\frac{4}{2}=\frac{y - 3}{x}\) to \(2=\frac{y - 3}{x}\). Multiply both sides by \(x\): \(2x=y - 3\). Add 3 to both sides: \(y = 2x+3\).

Answer:

The correct option is the fourth one (the one with \(\frac{BD}{DA}=\frac{CE}{EA}\) so \(\frac{4}{2}=\frac{y - 3}{x}\); solve for \(y\) to get \(y = 2x + 3\)).