HardMathematicsHigh School (11–12)Vectors and Three-Dimensional Geometry › Introduction to Three-Dimensional GeometryNumerical
A designer plans a square tile in space…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A designer plans a square tile in space. Three corners of the quadrilateral , taken in order round the tile, are , and , with distances in decimetres. The fourth corner makes a parallelogram. The designer thinks that it is a square. Find the three coordinates of . Then test the claim: compare the squared sides and the squared diagonals. Last give the area of the tile in square decimetres, using the side length.
Need a reminder? The key ideaShow
The centroid of a triangle is the mean of its 3 corners, and of a tetrahedron the mean of its 4. A parallelogram has . Points are collinear if or if one section ratio fits all three coordinates.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The fourth corner. .
Step 2. The squared sides. and . Adjacent sides are equal.
Step 3. The squared diagonals. and . The diagonals are equal.
Step 4. The shape and the area. A parallelogram with equal adjacent sides and equal diagonals is a square. Its side is , so the area is .
Check. , so each diagonal squared is twice the side squared, as in every square.
Answer: and the area is .
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