Find the lengths of the medians of the triangle with vertices
A(0, 0, 6), B(0, 4, 0) and C(6, 0, 0).
Step 1: Find the midpoints of the sides
Midpoint of BC:
\( M_1 = \left(\frac{0+6}{2}, \frac{4+0}{2}, \frac{0+0}{2}\right) = (3, 2, 0) \)
Midpoint of AC:
\( M_2 = \left(\frac{0+6}{2}, \frac{0+0}{2}, \frac{6+0}{2}\right) = (3, 0, 3) \)
Midpoint of AB:
\( M_3 = \left(\frac{0+0}{2}, \frac{0+4}{2}, \frac{6+0}{2}\right) = (0, 2, 3) \)
Step 2: Find the lengths of the medians
Median from A to BC:
\( AM_1 = \sqrt{(3-0)^2 + (2-0)^2 + (0-6)^2} \)
\( = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \)
Median from B to AC:
\( BM_2 = \sqrt{(3-0)^2 + (0-4)^2 + (3-0)^2} \)
\( = \sqrt{9 + 16 + 9} = \sqrt{34} \)
Median from C to AB:
\( CM_3 = \sqrt{(0-6)^2 + (2-0)^2 + (3-0)^2} \)
\( = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \)
Final Answer:
Lengths of the medians are
7, \( \sqrt{34} \), and 7.