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. statement: slope cannot be a ratio., definition of slope, slope from p to q = \\(\frac{e}{f}\\), f - e = f - e, \\(\frac{e}{f}=\frac{e}{f}\\), slope from q to r = \\(\frac{e}{f}\\). reason: definition of slope. triangle 1 is similar to triangle 2.

Explanation:

Step1: Recall slope definition

The slope between two points \((x_1,y_1)\) and \((x_2,y_2)\) is defined as \(m=\frac{y_2 - y_1}{x_2 - x_1}\), which is a ratio. So the statement "Slope cannot be a ratio" is incorrect and should not be used.

Step2: Identify correct slope - statements

The slope from \(P\) to \(Q=\frac{E}{F}\) and slope from \(Q\) to \(R=\frac{E'}{F'}\) are correct expressions based on the rise - over - run definition of slope for the right - angled triangles formed.

Step3: Use similarity of triangles

Since triangle 1 is similar to triangle 2, the ratios of corresponding sides are equal. That is, \(\frac{E}{F}=\frac{E'}{F'}\) because for similar right - angled triangles (formed by the line and the coordinate - axis), the ratio of the vertical side to the horizontal side (which is the slope) is the same. Also, if we consider the change in \(y\) and change in \(x\) values for the two segments of the line, the equality of these ratios is a result of the similarity of the triangles. And the reason for the equality of slopes is the definition of similar triangles (corresponding sides of similar triangles are in proportion).

Answer:

Statement: Slope from \(P\) to \(Q=\frac{E}{F}\), Reason: Definition of slope; Statement: Slope from \(Q\) to \(R=\frac{E'}{F'}\), Reason: Definition of slope; Statement: \(\frac{E}{F}=\frac{E'}{F'}\), Reason: Definition of similar triangles (corresponding sides of similar triangles are in proportion)