Solve this question---**17. (a)** * **Question Stem:** It is given that WXY is a triangle, where WX = 6 cm, XY = 5 cm and ∠WYX = 70°. Find ∠XWY. * **Marks:** (2 marks) **17. (b)** * **Problem Description:** Figure 3 shows the pyramid WXYZ, where WZ = XZ = YZ. The base of this pyramid is the triangle WXY described in (a). * **Chart/Diagram Description:** * **Type:** 3D geometric figure, specifically a pyramid. * **Title/Label:** Figure 3 * **Main Elements:** * **Vertices:** Four vertices labeled W, X, Y, and Z. * **Base:** The base of the pyramid is triangle WXY. The edges WX and XY are drawn as solid lines. The edge YW (or WY) is drawn as a dashed line, indicating it is a hidden or rear edge from the viewer's perspective. * **Apex:** Vertex Z is positioned above the base WXY. * **Slant Edges:** Three edges connect the apex Z to the base vertices: ZW, ZX, and ZY. All are drawn as solid lines. * **Question Stem (continued):** It is given that the angle between WZ and the triangle WXY is 30°. Does the angle between the triangles WXY and XYZ exceed 45°? Explain your answer. * **Marks:** (4 marks)

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We have triangle WXY with side WX equal to 6 centimeters, side XY equal to 5 centimeters, and angle WYX equal to 70 degrees. We need to find angle XWY. This is a classic triangle problem that can be solved using the Law of Sines. We apply the Law of Sines which states that the ratio of each side to the sine of its opposite angle is constant. For our triangle, XY over sine of angle XWY equals WX over sine of angle WYX. Substituting our values: 5 over sine of angle XWY equals 6 over sine of 70 degrees. Solving for sine of angle XWY, we get 5 times sine of 70 degrees divided by 6, which equals approximately 0.783. Taking the inverse sine, angle XWY is approximately 51.4 degrees. We start with triangle WXY where WX equals 6 centimeters, XY equals 5 centimeters, and angle WYX equals 70 degrees. We need to find angle XWY. Using the Law of Sines, we can set up the equation: WX over sine of angle WYX equals XY over sine of angle XWY. Substituting our values gives us 6 over sine of 70 degrees equals 5 over sine of angle XWY. Solving for sine of angle XWY, we get 5 times sine of 70 degrees divided by 6, which equals approximately 0.782. Therefore, angle XWY is approximately 51.3 degrees. Now we construct pyramid WXYZ using triangle WXY as the base. The apex Z is positioned such that WZ equals XZ equals YZ, making this a right pyramid. This means Z is directly above the circumcenter H of triangle WXY. The key information is that the angle between line WZ and the base plane is 30 degrees. We can use this to find the height of the pyramid, which equals the distance from H to W times the tangent of 30 degrees. To determine if the dihedral angle between planes WXY and XYZ exceeds 45 degrees, we analyze the geometry of our pyramid. The dihedral angle is the angle between the two planes along their common edge XY, shown in yellow. The blue plane represents the base WXY, and the red plane represents triangle XYZ. Given that Z is positioned directly above the circumcenter with a 30-degree angle to the base, the dihedral angle between these planes is approximately 30 degrees, which is clearly less than 45 degrees. Therefore, the answer is NO - the dihedral angle does not exceed 45 degrees. To find the pyramid's height, we first calculate the circumradius of triangle WXY. Using the formula R equals a times b times c divided by 4 times the area, we need to find the area and the third side WY. The angle at X is 180 minus 70 minus 51.3, which equals 58.7 degrees. The area is half times 6 times 5 times sine of 58.7 degrees, approximately 12.8 square centimeters. Using the law of cosines, WY equals approximately 6.4 centimeters. Therefore, the circumradius R equals 6 times 5 times 6.4 divided by 4 times 12.8, which is approximately 3.75 centimeters. The height of the pyramid is R times tangent of 30 degrees, which equals 3.75 times 1 over square root of 3, approximately 2.17 centimeters. To find the dihedral angle between planes WXY and XYZ, we need to measure the angle along their common edge XY, shown in yellow. First, we find the distance from the circumcenter H to edge XY, which is approximately 2.1 centimeters. The dihedral angle theta can be calculated using the tangent function: tangent of theta equals the height h divided by this distance d. So tangent of theta equals 2.17 divided by 2.1, which is approximately 1.03. Taking the inverse tangent, we get theta equals approximately 46 degrees. Since 46 degrees is greater than 45 degrees, the answer is YES - the dihedral angle between planes WXY and XYZ does exceed 45 degrees.