The cube in the figure above has edges of length 6 If point p is the midpoint of an edge, what is the perimeter of `/_\QPR?`

Options

A. 13.41
B. 14.28
C. 17.1
D. 23.81

Explanation

To find the perimeter of triangle QPR in the given cube, we will follow these steps:

### Step 1: Identify the Lengths of the Edges
The edge length of the cube is given as 6 units.

### Step 2: Determine the Position of Point P
Point P is the midpoint of an edge. Therefore, the length of segment PM (where M is one endpoint of the edge) and PN (where N is the other endpoint) will each be half of the edge length:
\[ PM = PN = \frac{6}{2} = 3 \text{ units} \]

### Step 3: Calculate the Length of QR
QR is the diagonal of the cube. The formula for the diagonal \(d\) of a cube with edge length \(A\) is:
\[ d = \sqrt{3A^2} \]
Substituting \(A = 6\):
\[ QR = \sqrt{3 \times 6^2} = \sqrt{3 \times 36} = \sqrt{108} = 6\sqrt{3} \text{ units} \]

### Step 4: Calculate the Length of PQ
To find PQ, we will use the right triangle PQN where:
- QN = 6 (the edge of the cube)
- PN = 3 (the midpoint)

Using the Pythagorean theorem:
\[ PQ^2 = QN^2 + PN^2 \]
\[ PQ^2 = 6^2 + 3^2 = 36 + 9 = 45 \]
Thus,
\[ PQ = \sqrt{45} = 3\sqrt{5} \text{ units} \]

### Step 5: Calculate the Length of PR
Triangle PMR is similar to triangle PQN, and thus we can use the same calculations:
\[ PR^2 = PM^2 + MR^2 \]
Where:
- PM = 3 (the midpoint)
- MR = 6 (the edge of the cube)

Using the Pythagorean theorem:
\[ PR^2 = 3^2 + 6^2 = 9 + 36 = 45 \]
Thus,
\[ PR = \sqrt{45} = 3\sqrt{5} \text{ units} \]

### Step 6: Calculate the Perimeter of Triangle QPR
The perimeter \(P\) of triangle QPR is given by:
\[ P = PQ + QR + PR \]
Substituting the values we found:
\[ P = 3\sqrt{5} + 6\sqrt{3} + 3\sqrt{5} \]
\[ P = 6\sqrt{5} + 6\sqrt{3} \]

### Step 7: Final Calculation
To approximate the values:
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{3} \approx 1.732 \)

Calculating:
\[ P \approx 6(2.236) + 6(1.732) \]
\[ P \approx 13.416 + 10.392 = 23.808 \]

Thus, the perimeter of triangle QPR is approximately \(23.81\) units.

### Final Answer
The perimeter of triangle QPR is \(23.81\) units.

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