To understand how similar triangles can help prove that the slope between points A and B is the same as the slope between A and C, I’ll start by recalling what slope is. The slope between two points (x₁, y₁) and (x₂, y₂) is defined as the change in y over the change in x, or mathematically, \( \text{slope} = \frac{y₂ - y₁}{x₂ - x₁} \).
Let’s say I have three points: A (x₁, y₁), B (x₂, y₂), and C (x₃, y₃). I want to show that the slope between points A and B (let’s call it \( m_{AB} \)) is the same as the slope between points A and C (let’s call it \( m_{AC} \)).
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Setting Up the Triangles: I can plot points A, B, and C on a coordinate grid. By drawing the straight lines connecting A to B and A to C, I can see that they create two triangles with the horizontal axis (the x-axis). The vertical lines dropped from B and C to the x-axis form the heights of the triangles, and the horizontal line connecting where those vertical lines meet the x-axis gives me the bases of the triangles.
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Identifying the Similar Triangles: If I visualize the right triangles formed, I can see that both triangles share the base, which is the distance between the projections of B and C on the x-axis. The height of the triangle with point B will be \( y₂ - y₁ \) (the vertical rise from A to B), while the height of the triangle with point C will be \( y₃ - y₁ \) (the vertical rise from A to C).
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Using the Property of Similarity: Since both triangles share an angle at point A (the angle between the line AB and the horizontal x-axis is the same as the angle between the line AC and the x-axis), and they both have a right angle, the triangles are similar by the AA (Angle-Angle) similarity postulate.
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Comparing the Ratios: Because the triangles are similar, the ratios of their corresponding sides must be equal. This gives me the following relation:
\[ \frac{\text{rise between A and B}}{\text{run between A and B}} = \frac{\text{rise between A and C}}{\text{run between A and C}} \]
Expressing this mathematically, I get:
\[ \frac{y₂ - y₁}{x₂ - x₁} = \frac{y₃ - y₁}{x₃ - x₁} \]
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Concluding: Therefore, I can conclude that the slope \( m_{AB} \) (between A and B) is indeed equal to the slope \( m_{AC} \) (between A and C), which shows that these slopes are the same due to the similarity of the triangles formed by the points.
By understanding these concepts, I can confidently explain how similar triangles help prove that the slopes of two different lines from a common point are equal!