This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's the complete solution: Step 1: Find the coordinates of point A. Point A is the y-intercept of the line y = 2x + 3. To find the y-intercept, we set x = 0. y = 2(0) + 3 y = 3 The coordinates of point A are (0, 3). Step 2: Understand the relationship between A, B, and C. Points B and C are on the line y = 2x + 3. The condition AB = AC means that point A is the midpoint of the line segment BC, because A, B, and C are collinear. Step 3: Find the slope of line AC. The line AC is the line y = 2x + 3. The slope of this line, m_AC, is the coefficient of x. m_AC = 2 Step 4: Find the slope of line BD. Line BD is perpendicular to line AC. The product of the slopes of two perpendicular lines is -1. Let m_BD be the slope of line BD. m_BD · m_AC = -1 m_BD · 2 = -1 m_BD = -(1)/(2) Step 5: Find the equation of line BD. Line BD passes through point D(1, 0) and has a slope m_BD = -(1)/(2). Using the point-slope form y - y_1 = m(x - x_1): y - 0 = -(1)/(2)(x - 1) y = -(1)/(2)x + (1)/(2) The equation of BD is y = -(1)/(2)x + (1)/(2). Step 6: Find the coordinates of point B. Point B is the intersection of the line y = 2x + 3 and the line y = -(1)/(2)x + (1)/(2). We set the y-values equal to find the x-coordinate: 2x + 3 = -(1)/(2)x + (1)/(2) Multiply the entire equation by 2 to clear fractions: 2(2x + 3) = 2(-(1)/(2)x + (1)/(2)) 4x + 6 = -x + 1 5x = -5 x = -1 Substitute x = -1 into the equation y = 2x + 3 to find the y-coordinate: y = 2(-1) + 3 y = -2 + 3 y = 1 The coordinates of point B are (-1, 1). Step 7: Find the coordinates of point C. Since A(0, 3) is the midpoint of BC, and B is (-1, 1), we use the midpoint formula: x_A = (x_B + x_C)/(2) and y_A = (y_B + y_C)/(2) For the x-coordinate: 0 = (-1 + x_C)/(2) 0 = -1 + x_C x_C = 1 For the y-coordinate: 3 = (1 + y_C)/(2) 6 = 1 + y_C y_C = 5 The coordinates of point C are (1, 5). That's 2 down. 3 left today — send the next one.