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The 3D graphic shows a trophy which is to be awarded to the best student in a DGC class - Leaving Cert DCG - Question B-2 - 2009
Question B-2
The 3D graphic shows a trophy which is to be awarded to the best student in a DGC class. The trophy consists of two interlocking coloured glass set squares. (The ma... show full transcript
Worked Solution & Example Answer:The 3D graphic shows a trophy which is to be awarded to the best student in a DGC class - Leaving Cert DCG - Question B-2 - 2009
Step 1
Draw the plan and elevation of the intersecting planes.
Answer
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Start by plotting the points A, B, and C based on their coordinates:
- Point A (145, 80, 30)
- Point B (266, 80, 30)
- Point C (170, 80, 75)
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Connect points A, B, and C to form triangle ABC.
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Similarly, plot points D, E, and F:
- Point D (250, 90, 30)
- Point E (145, 90, 30)
- Point F (250, 90, 75)
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Connect points D, E, and F to create triangle DEF.
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Now, project the positions of triangles ABC and DEF to create a plan view and an elevation view on the same drawing.
Ensure appropriate line types are used for hidden and visible edges.
Step 2
Determine the line of intersection between the planes.
Answer
To find the intersection line of the planes formed by triangles ABC and DEF:
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Identify the line segments of both triangles that intersect.
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Use vector equations to determine the intersection points. For example, using the coordinates of points A and D, and checking their directional vectors.
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The line of intersection lies along the coordinates that satisfy both plane equations at the points where triangle edges intersect.
Step 3
Determine the dihedral angle between the planes.
Answer
The dihedral angle between two planes can be found using the normal vectors:
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Determine the normal vectors of triangles ABC and DEF.
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For triangle ABC, calculate using the cross product of two vectors formed by its sides:
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For triangle DEF, apply the same method:
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Use the dot product to find the angle: ext{cos}( heta) = rac{ ext{Normal}_{ABC} ullet ext{Normal}_{DEF}}{| ext{Normal}_{ABC}| | ext{Normal}_{DEF}|}
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Calculate the angle using:
Step 4
Draw the plan and elevation of the inner triangle.
Answer
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Offset triangle ABC by 10mm towards the inside of the 45° set square:
- Adjust the coordinates of points A, B, and C accordingly.
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Plot the new coordinates and connect the points to form the inner triangle.
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Draw the plan and elevation of this inner triangle clearly distinguishing it from the outer triangle.
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Ensure all dimensions are labeled appropriately.