A line passes through the points \( (3, 7) \) and \( (-1, -5) \). What is the slope of a line perpendicular to this line?
["SEO Optimized Article:\nHow to Calculate the Slope of a Perpendicular Line to the Line Passing Through (3, 7) and (-1, -5)", "When learning geometry, one essential concept is finding the slope of a line and how to determine a perpendicular slope. In this article, we’ll walk through step-by-step how to find the slope of a line passing through two given points — specifically, the points ( (3, 7) ) and ( (-1, -5) ) — and then calculate the slope of a line perpendicular to it.", "### Step 1: Find the Slope of the Given Line", "The formula for the slope ( m ) of a line passing through two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is:\n[\nm = \frac{y_2 - y_1}{x_2 - x_1}\n]", "Using the points ( (3, 7) ) and ( (-1, -5) ):\n- ( x_1 = 3, y_1 = 7 )\n- ( x_2 = -1, y_2 = -5 )", "Substitute into the formula:\n[\nm = \frac{-5 - 7}{-1 - 3} = \frac{-12}{-4} = 3\n]", "So, the slope of the original line is ( 3 ).", "### Step 2: Find the Slope of a Perpendicular Line", "A key rule in geometry is that slopes of perpendicular lines are negative reciprocals of each other. If the original slope is ( m ), then the perpendicular slope ( m_{\perp} ) is:\n[\nm_{\perp} = -\frac{1}{m}\n]", "Since ( m = 3 ):\n[\nm_{\perp} = -\frac{1}{3}\n]", "### Summary", "- The slope of the line through ( (3, 7) ) and ( (-1, -5) ) is ( 3 ).\n- The slope of any line perpendicular to it is ( -\frac{1}{3} ).", "Understanding these concepts helps in solving problems involving parallel and perpendicular lines, which are fundamental in coordinate geometry. Whether you're preparing for school exams or enhancing your math skills, mastering slope and perpendicularity is key.", "Keywords for SEO:\nslope calculation, perpendicular line slope, line through points (3, 7) and (-1, -5), negative reciprocal slope, geometry tutorial, slope and perpendicular slopes, coordinate geometry, math problem solving", "Meta Description:\nLearn how to find the slope of a line passing through points (3, 7) and (-1, -5), and discover how to determine the slope of a line perpendicular to it using geometry fundamentals. Perfect for students and math learners.", "---", "This article explains clearly, uses structured heading tags (H2, H3), and incorporates relevant keywords naturally—ideal for SEO while remaining accessible and useful."]
