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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.
Definition of slope
F – E = F’ – E’
Slope cannot be a ratio.
E’ / E = F / E
Slope from P to Q = F / E
To complete Ethan’s proof about the slope between any two points on a straight line, we can use the following steps:
1. Definition of Slope: The slope (m) between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line is defined as ( m = frac{y_2 – y_1}{x_2 – x_1} ).
2. Using Similar Triangles: Since Ethan has already proved that triangles 1 and 2 are similar, we can use the property of similar triangles which states that the ratios of corresponding sides are equal.
3. Finding the slopes of the triangles: If we denote the lengths of the sides as follows: ( F’ ) for the vertical distance (rise) and ( E’ ) for the horizontal distance (run) for triangle 1, and ( F ) for the vertical distance and ( E ) for the horizontal distance for triangle 2, we have:
[
frac{F’}{E’} = frac{F}{E}
]
4. Conclusion: Since the triangles are similar, the slopes from point P to Q will be consistent, meaning:
[
text{Slope from P to Q} = frac{F}{E} text{ and } text{Slope from any point } (E, F) text{ to another point