EasyMathematicsHigh School (11–12)Vectors and Three-Dimensional Geometry › Introduction to Three-Dimensional GeometryNumerical
Three sensors are fixed on a sloping roof at A(3, 0, 6), B(0, 6, 3) and C(6, 3, 0)…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
Three sensors are fixed on a sloping roof at , and . Distances are in metres from a reference point at the origin. A single cable box is to be placed at the centroid of the triangle . Find the coordinates of the centroid, and then its distance from the origin in metres, to two decimal places.
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 centroid. , and . So .
Step 2. The distance. .
Check. The point lies on the line through the origin, which is the symmetry line of the three corners.
Answer: the centroid is and is from the origin.
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