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Given below are descriptions of two lines. Line 1: Goes through \( (12,-16) \) and \( (11,-14) \) Line 2: Goes through \( (4,-29) \) and \( (9,-74) \) The slope of Line 1 is \( m= \) The slope of Line 2 is \( m= \) Finally, which of the following is true? Line 1 is parallel to Line 2. Line 1 is perpendicular to Line 2 Line 1 is neither parallel nor perpendicular to Line 2 Question Help: Submit Question 1 Video 1 Video 2 Video 3

Ask by Campos Mitchell. in the United States
Mar 14,2025

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The slope of Line 1 is \( m_1 = -2 \) and the slope of Line 2 is \( m_2 = -9 \). Therefore, Line 1 is neither parallel nor perpendicular to Line 2.

Solución

To find the slopes of the two lines, we can use the formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] ### Step 1: Calculate the slope of Line 1 For Line 1, the points are \( (12, -16) \) and \( (11, -14) \). Using the slope formula: \[ m_1 = \frac{-14 - (-16)}{11 - 12} \] Calculating the numerator and denominator: \[ m_1 = \frac{-14 + 16}{11 - 12} = \frac{2}{-1} = -2 \] ### Step 2: Calculate the slope of Line 2 For Line 2, the points are \( (4, -29) \) and \( (9, -74) \). Using the slope formula: \[ m_2 = \frac{-74 - (-29)}{9 - 4} \] Calculating the numerator and denominator: \[ m_2 = \frac{-74 + 29}{9 - 4} = \frac{-45}{5} = -9 \] ### Step 3: Determine the relationship between the lines Now we have the slopes: - Slope of Line 1: \( m_1 = -2 \) - Slope of Line 2: \( m_2 = -9 \) To determine if the lines are parallel, perpendicular, or neither: - Lines are parallel if their slopes are equal: \( m_1 = m_2 \) - Lines are perpendicular if the product of their slopes is \(-1\): \( m_1 \cdot m_2 = -1 \) Calculating the product of the slopes: \[ m_1 \cdot m_2 = (-2) \cdot (-9) = 18 \] Since \( 18 \neq -1 \) and \( -2 \neq -9 \), we conclude that: - Line 1 is neither parallel nor perpendicular to Line 2. ### Final Results - The slope of Line 1 is \( m_1 = -2 \). - The slope of Line 2 is \( m_2 = -9 \). - Therefore, the correct statement is: **Line 1 is neither parallel nor perpendicular to Line 2.**

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Mind Expander

To find the slopes of the lines, we can use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For Line 1, with points \( (12, -16) \) and \( (11, -14) \): \[ m_1 = \frac{-14 - (-16)}{11 - 12} = \frac{2}{-1} = -2 \] For Line 2, with points \( (4, -29) \) and \( (9, -74) \): \[ m_2 = \frac{-74 - (-29)}{9 - 4} = \frac{-45}{5} = -9 \] Now, we see that Line 1 has a slope of \(-2\) and Line 2 has a slope of \(-9\). To determine their relationship, we check if the slopes are equal (parallel) or if the product of their slopes equals \(-1\) (perpendicular). Since \(-2\) is not equal to \(-9\) and \(-2 \times -9 = 18 \neq -1\), we conclude: Line 1 is neither parallel nor perpendicular to Line 2. You know, the beauty of slopes is that they tell a tale of angles and inclination! Parallel lines will always dance side by side, but perpendicular ones cross paths at that perfect right angle, like the ultimate intersection of roads! For a practical application, think about graphing your own equations or even designing something with right angles and slopes, such as architectural blueprints or video game levels. Understanding how lines relate to each other can help in visualizing and solving real-world scenarios!

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