Question

ethan is proving that the slope between any two points on a straight line is the same. he has already proved that triangles 1 and 2 are similar. drag statements and reasons to complete the proof. slope cannot be a ratio. definition of slope slope from p to q = \\(\frac{f}{e}\\) f - e = f - e \\(\frac{f}{e}=\frac{f}{e}\\) slope from q to r = \\(\frac{f}{e}\\) reason: definition of slope

Explanation:

Step1: Recall the definition of slope

The slope between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For the slope from \(P\) to \(Q\) and from \(Q\) to \(R\), we use the properties of similar triangles. Since triangles \(1\) and \(2\) are similar, the ratios of their corresponding sides are equal.

Step2: Use the property of similar - triangles

If two triangles are similar, then \(\frac{E}{F}=\frac{E'}{F'}\) (corresponding - side - ratio property of similar triangles).

Step3: Apply the slope - formula and similar - triangle property

The slope from \(P\) to \(Q\) is \(\frac{E}{F}\) (by the definition of slope, where \(E\) is the vertical change and \(F\) is the horizontal change) and the slope from \(Q\) to \(R\) is \(\frac{E'}{F'}\). Since \(\frac{E}{F}=\frac{E'}{F'}\) (because of similar triangles), and the slope from \(P\) to \(Q\) and from \(Q\) to \(R\) are calculated using the ratio of vertical - change to horizontal - change (definition of slope).

Answer:

Statement: Slope from \(Q\) to \(R=\frac{E'}{F'}\)
Reason: \(\frac{E}{F}=\frac{E'}{F'}\)