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The 3D graphic on the right shows a metal geometric sculpture made up of a series of intersecting triangles - Leaving Cert DCG - Question B-3 - 2010
Question B-3
The 3D graphic on the right shows a metal geometric sculpture made up of a series of intersecting triangles. The horizontal and vertical coordinates for the outer p... show full transcript
Worked Solution & Example Answer:The 3D graphic on the right shows a metal geometric sculpture made up of a series of intersecting triangles - Leaving Cert DCG - Question B-3 - 2010
Step 1
Draw the plan and elevation of the intersecting planes.
Answer
To draw the plan and elevation of the intersecting triangular planes ABC and ABD, start by plotting the points A, B, C, and D on plan and elevation views using the provided coordinates:
- Point A: (60, 70)
- Point B: (90, 40)
- Point C: (120, 15)
- Point D: (30, 70)
Next, connect the points according to the triangular compositions:
- For triangle ABC, connect A to B to C.
- For triangle ABD, connect A to B to D.
Ensure that the angles are properly adjusted to reflect their three-dimensional orientation.
Step 2
Determine the dihedral angle between the planes.
Answer
To determine the dihedral angle between planes ABC and ABD, utilize the normal vectors of each plane. The dihedral angle θ can be calculated using the formula:
heta = ext{acos} rac{n_1 ullet n_2}{|n_1| |n_2|}
where and are the normal vectors of the planes. After calculating the normal vectors through cross products of the edges of each triangle, substitute the values and evaluate the angle.
Step 3
Determine the true shape of the triangle ABC.
Answer
To find the true shape of triangle ABC, project the triangle onto a plane parallel to it. This can be done by determining the angles and lengths of each side using trigonometric relations. Utilize the coordinates to find actual measurements, which can be represented in a true scale-drawing.
Step 4
Determine the traces of the plane that contains the triangle ABC.
Answer
The traces of the plane can be determined by finding the intersection points with the coordinate planes (XY, XZ, YZ). This involves creating equations for the plane using the coordinates A, B, and C and then solving for where the plane intersects each of the coordinate axes to find the traces.