Question

the cuboid, oabcdefg is such that oa = 8, oc = 9, and od = 4. a lies on the x - axis, c lies on the y - axis and d lies on the z - axis. 1. find the coordinates of point b 2. find the coordinates of point d 3. find the distance from b to d 4. find the midpoint between points b and d

Explanation:

Step1: Determine coordinates of B

Since A lies on x - axis with OA = 8, C lies on y - axis with OC = 9 and B is in the x - y plane of the cuboid, the coordinates of B are (8,9,0).

Step2: Determine coordinates of D

Given D lies on z - axis with OD = 4, the coordinates of D are (0,0,4).

Step3: Calculate distance between B and D

Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2+(z_2 - z_1)^2}$. Here, $(x_1,y_1,z_1)=(8,9,0)$ and $(x_2,y_2,z_2)=(0,0,4)$. So $d=\sqrt{(0 - 8)^2+(0 - 9)^2+(4 - 0)^2}=\sqrt{64 + 81+16}=\sqrt{161}$.

Step4: Find mid - point between B and D

Use the mid - point formula $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2},\frac{z_1 + z_2}{2})$. Here, $(x_1,y_1,z_1)=(8,9,0)$ and $(x_2,y_2,z_2)=(0,0,4)$. So $M=(\frac{8 + 0}{2},\frac{9+0}{2},\frac{0 + 4}{2})=(4,\frac{9}{2},2)$.

Answer:

1. (8,9,0)
2. (0,0,4)
3. $\sqrt{161}$
4. $(4,\frac{9}{2},2)$