Let `O(0,0,0),P(3,4,5),Q(m,n,r) and R(1,1,1)` be the vertices of a parallelogram taken in order. What is the value of m+n+r?

Options

A. 6
B. 12
C. 15
D. 8

Answer

To solve the problem, we need to find the coordinates of point Q (m, n, r) such that O, P, Q, and R form a parallelogram. The coordinates of the points are given as follows:

- O(0, 0, 0)
- P(3, 4, 5)
- R(1, 1, 1)
- Q(m, n, r)

### Step 1: Understand the properties of a parallelogram
In a parallelogram, the diagonals bisect each other. This means that the midpoint of diagonal OP should be the same as the midpoint of diagonal QR.

### Step 2: Calculate the midpoint of OP
The midpoint of OP can be calculated using the midpoint formula:
\[ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \]
For points O(0, 0, 0) and P(3, 4, 5):
\[ \text{Midpoint of OP} = \left(\frac{0 + 3}{2}, \frac{0 + 4}{2}, \frac{0 + 5}{2}\right) = \left(\frac{3}{2}, 2, \frac{5}{2}\right) \]

### Step 3: Calculate the midpoint of QR
Now, we need to find the midpoint of QR, where Q is (m, n, r) and R is (1, 1, 1):
\[ \text{Midpoint of QR} = \left(\frac{m + 1}{2}, \frac{n + 1}{2}, \frac{r + 1}{2}\right) \]

### Step 4: Set the midpoints equal to each other
Since the midpoints of the diagonals must be equal, we can set the coordinates equal to each other:
\[ \frac{m + 1}{2} = \frac{3}{2} \quad (1) \]
\[ \frac{n + 1}{2} = 2 \quad (2) \]
\[ \frac{r + 1}{2} = \frac{5}{2} \quad (3) \]

### Step 5: Solve the equations for m, n, and r
From equation (1):
\[ m + 1 = 3 \implies m = 2 \]

From equation (2):
\[ n + 1 = 4 \implies n = 3 \]

From equation (3):
\[ r + 1 = 5 \implies r = 4 \]

### Step 6: Calculate m + n + r
Now we can find the value of \( m + n + r \):
\[ m + n + r = 2 + 3 + 4 = 9 \]

### Final Answer
Thus, the value of \( m + n + r \) is **9**.

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