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
4. (a) Illustrate the information in a diagram. A diagram showing the journey: Start at point E. Draw a North line vertically upwards from E. From E, draw a line segment EF, 4 km long, at an angle of 135^ clockwise from the North line. Point F is the end of this segment. From point F, draw another North line vertically upwards, parallel to the first North line. From F, draw a line segment FG, 13 km long, at an angle of 045^ clockwise from the North line at F. Point G is the end of this segment. Connect points E and G to form triangle EFG. The angle EFG can be determined from the bearings. The bearing of F from E is 135^. The angle between the North line at F and the line FE (pointing towards E) is 180^ - 135^ = 45^ (alternate interior angles with the 135^ bearing from E, considering the line from F to E). The bearing of G from F is 045^. Therefore, the interior angle EFG = 45^ + 45^ = 90^. 4. (b) i. Find |EG| correct to two significant figures. Step 1: Identify the sides and angle of triangle EFG. We have |EF| = 4 km, |FG| = 13 km, and EFG = 90^. Step 2: Use the cosine rule to find |EG|. Since EFG = 90^, we can use the Pythagorean theorem. |EG|^2 = |EF|^2 + |FG|^2 - 2|EF||FG|( EFG) |EG|^2 = 4^2 + 13^2 - 2(4)(13)(90^) |EG|^2 = 16 + 169 - 2(4)(13)(0) |EG|^2 = 185 Step 3: Calculate |EG| and round to two significant figures. |EG| = sqrt(185) ≈ 13.60147 Correct to two significant figures, |EG| is 14 km. 4. (b) ii. The bearing of G from F. Step 1: Read the information directly from the problem statement. The problem states: "He continues 13km on a bearing of 045° to G." This means the bearing of G from F is 045^. The bearing of G from F is 045^. 5. (a) The midpoint of the line joining Y(2a, 3) and Z(-4, b) is M(1, 2a-1). Find the values of a and b. Step 1: Use the midpoint formula for the x-coordinate. The x-coordinate of the midpoint is M_x = (x_Y + x_Z)/(2). 1 = (2a + (-4))/(2) 2 = 2a - 4 6 = 2a a = 3 Step 2: Use the midpoint formula for the y-coordinate. The y-coordinate of the midpoint is M_y = (y_Y + y_Z)/(2). 2a - 1 = (3 + b)/(2) Step 3: Substitute the value of a found in Step 1 into the equation from Step 2 and solve for b. 2(3) - 1 = (3 + b)/(2) 6 - 1 = (3 + b)/(2) 5 = (3 + b)/(2) 10 = 3 + b b = 7 The values are a = 3, b = 7. 5. (b) Find |YZ|. Step 1: Determine the coordinates of points Y and Z using the values of a and b. Y(2a, 3) = Y(2(3), 3) = Y(6, 3) $Z