Given points A(-3, 2), C(9, 10), and D(4, -2). Calculate the gradient of AC, find the equation of DE (parallel to CA), and determine the size of angle alpha (formed by AC and the y-axis).

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Asked on May 12, 2026Mathematics

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.

Accepted Answer · 2026-05-12T22:27:17.802Z

Here are the solutions to the questions: Given points: A(-3, 2) C(9, 10) D(4, -2) 2.1 Calculate the gradient of the line AC. Step 1: Use the gradient formula m = (y_2 - y_1)/(x_2 - x_1) with points A(-3, 2) and C(9, 10). m_AC = (10 - 2)/(9 - (-3)) m_AC = (8)/(9 + 3) m_AC = (8)/(12) m_AC = (2)/(3) The gradient of line AC is (2)/(3). 2.2 Determine the equation of line DE in the form y = mx + c. Step 1: Since line DE is parallel to CA, their gradients are equal. m_DE = m_CA = (2)/(3) Step 2: Use the point-slope form y - y_1 = m(x - x_1) with point D(4, -2) and m_DE = (2)/(3). y - (-2) = (2)/(3)(x - 4) y + 2 = (2)/(3)x - (8)/(3) Step 3: Solve for y to get the equation in the form y = mx + c. y = (2)/(3)x - (8)/(3) - 2 y = (2)/(3)x - (8)/(3) - (6)/(3) y = (2)/(3)x - (14)/(3) The equation of line DE is y = (2)/(3)x - (14)/(3). 2.3 Calculate the size of . Step 1: Identify as CFO. F is the y-intercept of line AC. First, find the equation of line AC. We have m_AC = (2)/(3). Using point A(-3, 2): y - 2 = (2)/(3)(x - (-3)) y - 2 = (2)/(3)(x + 3) y - 2 = (2)/(3)x + 2 y = (2)/(3)x + 4 The y-intercept F is (0, 4). Step 2: Let be the angle of inclination of line AC with the positive x-axis. m_AC = (2)/(3) = = ((2)/(3)) ≈ 33.69^ Step 3: Relate to . From the diagram, is the angle between the line AC and the y-axis. In the right-angled triangle formed by the x-axis, y-axis, and line AC (let G be the x-intercept), and are complementary angles. = 90^ - = 90^ - 33.69^ = 56.31^ The size of is 56.3^. 2.4 B is a point in the first quadrant such that ABDE, in that order, forms a rectangle. Calculate, giving reasons, the: 2.4.1 Coordinates of M, the midpoint of BE. Step 1: In a rectangle, the diagonals bisect each other. Therefore, the midpoint of diagonal BE is the same as the midpoint of diagonal AD. Given points A(-3, 2) and D(4, -2). Step 2: Use the midpoint formula M = ((x_1 + x_2)/(2), (y_1 + y_2)/(2)) for AD. M = ((-3 + 4)/(2), (2 + (-2))/(2)) M = ((1)/(2), (0)/(2)) M = ((1)/(2), 0) The coordinates of M are ((1)/(2), 0). 2.4.2 Length of diagonal BE. Step 1: In a rectangle, the diagonals are equal in length. Therefore, the length of diagonal BE is equal to the length of diagonal AD. Given points A(-3, 2) and D(4, -2). Step 2: Use the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) for AD. BE = AD = sqrt((4 - (-3))^2 + (-2 - 2)^2) BE = sqrt((7)^2 + (-4)^2) BE = sqrt(49 + 16) BE = sqrt(65) The length of diagonal BE is sqrt(65). Drop the next question 📸

Accepted Answer · 2026-05-12T22:27:17.802Z

Here are the solutions to the questions: Given points: A(-3, 2) C(9, 10) D(4, -2) 2.1 Calculate the gradient of the line AC. Step 1: Use the gradient formula m = (y_2 - y_1)/(x_2 - x_1) with points A(-3, 2) and C(9, 10). m_AC = (10 - 2)/(9 - (-3)) m_AC = (8)/(9 + 3) m_AC = (8)/(12) m_AC = (2)/(3) The gradient of line AC is (2)/(3). 2.2 Determine the equation of line DE in the form y = mx + c. Step 1: Since line DE is parallel to CA, their gradients are equal. m_DE = m_CA = (2)/(3) Step 2: Use the point-slope form y - y_1 = m(x - x_1) with point D(4, -2) and m_DE = (2)/(3). y - (-2) = (2)/(3)(x - 4) y + 2 = (2)/(3)x - (8)/(3) Step 3: Solve for y to get the equation in the form y = mx + c. y = (2)/(3)x - (8)/(3) - 2 y = (2)/(3)x - (8)/(3) - (6)/(3) y = (2)/(3)x - (14)/(3) The equation of line DE is y = (2)/(3)x - (14)/(3). 2.3 Calculate the size of . Step 1: Identify as CFO. F is the y-intercept of line AC. First, find the equation of line AC. We have m_AC = (2)/(3). Using point A(-3, 2): y - 2 = (2)/(3)(x - (-3)) y - 2 = (2)/(3)(x + 3) y - 2 = (2)/(3)x + 2 y = (2)/(3)x + 4 The y-intercept F is (0, 4). Step 2: Let be the angle of inclination of line AC with the positive x-axis. m_AC = (2)/(3) = = ((2)/(3)) ≈ 33.69^ Step 3: Relate to . From the diagram, is the angle between the line AC and the y-axis. In the right-angled triangle formed by the x-axis, y-axis, and line AC (let G be the x-intercept), and are complementary angles. = 90^ - = 90^ - 33.69^ = 56.31^ The size of is 56.3^. 2.4 B is a point in the first quadrant such that ABDE, in that order, forms a rectangle. Calculate, giving reasons, the: 2.4.1 Coordinates of M, the midpoint of BE. Step 1: In a rectangle, the diagonals bisect each other. Therefore, the midpoint of diagonal BE is the same as the midpoint of diagonal AD. Given points A(-3, 2) and D(4, -2). Step 2: Use the midpoint formula M = ((x_1 + x_2)/(2), (y_1 + y_2)/(2)) for AD. M = ((-3 + 4)/(2), (2 + (-2))/(2)) M = ((1)/(2), (0)/(2)) M = ((1)/(2), 0) The coordinates of M are ((1)/(2), 0). 2.4.2 Length of diagonal BE. Step 1: In a rectangle, the diagonals are equal in length. Therefore, the length of diagonal BE is equal to the length of diagonal AD. Given points A(-3, 2) and D(4, -2). Step 2: Use the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) for AD. BE = AD = sqrt((4 - (-3))^2 + (-2 - 2)^2) BE = sqrt((7)^2 + (-4)^2) BE = sqrt(49 + 16) BE = sqrt(65) The length of diagonal BE is sqrt(65). Drop the next question 📸

Given points A(-3, 2), C(9, 10), and D(4, -2). Calculate the gradient of AC, find the equation of DE (parallel to CA), and determine the size of angle alpha (formed by AC and the y-axis).

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Given points A(-3, 2), C(9, 10), and D(4, -2). Calculate the gradient of AC, find the equation of DE (parallel to CA), and determine the size of angle alpha (formed by AC and the y-axis).

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