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
Welcome back NLE — been a while! Let's pick up where you left off. This problem involves plotting points and performing geometric transformations. Since I cannot draw graphs, I will provide the coordinates for each step. a) The coordinates of ABC are given as A(5,7), B(3,4), and C(7,3). You would plot these points on graph paper using the specified scale (2cm to 2 units on each axis) and connect them to form the triangle. b) To find the image A'B'C' under the translation by the vector 4 \\ -2 , we add the translation vector to each coordinate. Step 1: Calculate the coordinates of A'. A' = A + 4 \\ -2 = 5 \\ 7 + 4 \\ -2 = 5+4 \\ 7-2 = 9 \\ 5 So, A'(9,5). Step 2: Calculate the coordinates of B'. B' = B + 4 \\ -2 = 3 \\ 4 + 4 \\ -2 = 3+4 \\ 4-2 = 7 \\ 2 So, B'(7,2). Step 3: Calculate the coordinates of C'. C' = C + 4 \\ -2 = 7 \\ 3 + 4 \\ -2 = 7+4 \\ 3-2 = 11 \\ 1 So, C'(11,1). The image A'B'C' has coordinates A'(9,5), B'(7,2), and C'(11,1). c) To find the image A''B''C'' of ABC under a reflection in the line y = -2. For a reflection in the line y=k, a point (x,y) maps to (x, 2k-y). Here k=-2. Step 1: Calculate the coordinates of A''. A'' = (5, 2(-2)-7) = (5, -4-7) = (5, -11) So, A''(5,-11). Step 2: Calculate the coordinates of B''. B'' = (3, 2(-2)-4) = (3, -4-4) = (3, -8) So, B''(3,-8). Step 3: Calculate the coordinates of C''. C'' = (7, 2(-2)-3) = (7, -4-3) = (7, -7) So, C''(7,-7). The image A''B''C'' has coordinates A''(5,-11), B''(3,-8), and C''(7,-7). d) To find the image A'''B'''C''' of ABC under a rotation through 90^ anticlockwise about the origin. For a 90^ anticlockwise rotation about the origin, a point (x,y) maps to (-y,x). Step 1: Calculate the coordinates of A'''. A''' = (-7, 5) So, A'''(-7,5). Step 2: Calculate the coordinates of B'''. B''' = (-4, 3) So, B'''(-4,3). Step 3: Calculate the coordinates of C'''. C''' = (-3, 7) So, C'''(-3,7). The image A'''B'''C''' has coordinates A'''(-7,5), B'''(-4,3), and C'''(-3,7). e) Find the gradient of line B''B'''. From part (c), B''(3,-8). From part (d), B'''(-4,3). The formula for the gradient m of a line passing through (x_1, y_1) and (x_2, y_2) is m = (y_2 - y_1)/(x_2 - x_1). Step 1: Substitute the coordinates of B'' and B''' into the gradient formula. Let (x_1, y_1) = (3, -8) and (x_2, y_2) = (-4, 3). m = (3 - (-8))/(-4 - 3) Step 2: Simplify the expression. m = (3 + 8)/(-7) = (11)/(-7) = -(11)/(7) The gradient of line B''B''' is -(11)/(7). Drop the next question.