A line T passes through (5, 4) and (1, 2). a) Find the equation of T in the form y = mx + c. b) Find the equation of line D, the perpendicular bisector of line T in the form ax + by + c = 0.

You're on a roll — here are the solutions to the problems. 17. A line T passes through (5, 4) and (1, 2). a) Find the equation of T in the form y = mx + c, where m and c are constants. Step 1: Calculate the slope (m) of line T using the points (x_1, y_1) = (5, 4) and (x_2, y_2) = (1, 2). m = (y_2 - y_1)/(x_2 - x_1) = (2 - 4)/(1 - 5) = (-2)/(-4) = (1)/(2) Step 2: Use the point-slope form y - y_1 = m(x - x_1) with m = (1)/(2) and one of the points, for example (1, 2). y - 2 = (1)/(2)(x - 1) y - 2 = (1)/(2)x - (1)/(2) Step 3: Rearrange the equation into the form y = mx + c. y = (1)/(2)x - (1)/(2) + 2 y = (1)/(2)x + (4)/(2) - (1)/(2) y = (1)/(2)x + (3)/(2) The equation of line T is y = (1)/(2)x + (3)/(2). b) Find the equation of line D, the perpendicular bisector of line T. Leave your answer in the form ax + by + c = 0, where a, b and c are integers. Step 1: Find the midpoint of line T. The perpendicular bisector passes through the midpoint of the segment connecting (5, 4) and (1, 2). Midpoint = ((x_1+x_2)/(2), (y_1+y_2)/(2)) = ((5+1)/(2), (4+2)/(2)) = ((6)/(2), (6)/(2)) = (3, 3) Step 2: Find the slope of line D. Line D is perpendicular to line T. The slope of T is m_T = (1)/(2). The slope of a perpendicular line m_D is the negative reciprocal of m_T. m_D = -(1)/(m_T) = -(1)/(1/2) = -2 Step 3: Use the point-slope form y - y_M = m_D(x - x_M) with m_D = -2 and the midpoint (3, 3). y - 3 = -2(x - 3) y - 3 = -2x + 6 Step 4: Rearrange the equation into the form ax + by + c = 0. 2x + y - 3 - 6 = 0 2x + y - 9 = 0 The equation of line D is 2x + y - 9 = 0. c) Another line Y is parallel to line T at the point (-8, 0) and intersects with line D at point Z. Find the coordinates of Z. Step 1: Find the equation of line Y. Line Y is parallel to line T, so its slope m_Y is equal to m_T. m_Y = m_T = (1)/(2) Line Y passes through the point (-8, 0). Using the point-slope form: y - 0 = (1)/(2)(x - (-8)) y = (1)/(2)(x + 8) y = (1)/(2)x + 4 Step 2: Find the coordinates of Z by finding the intersection of line Y and line D. Equation of line Y: y = (1)/(2)x + 4 Equation of line D (from part b, rearranged to y=mx+c form): y = -2x + 9 Set the expressions for y equal to each other: (1)/(2)x + 4 = -2x + 9 Step 3: Solve for x. Multiply the entire equation by 2 to eliminate the fraction. 2((1)/(2)x + 4) = 2(-2x + 9) x + 8 = -4x + 18 x + 4x = 18 - 8 5x = 10 x = (10)/(5) x = 2 Step 4: Substitute the value of x into either equation (e.g., line Y) to find y. y = (1)/(2)(2) + 4 y = 1 + 4 y = 5 The coordinates of Z are (2, 5). What's next?