To find the numerical value of \( k \), we proceed as follows:

We know the sides of the equilateral triangle are \( 2p - q + 5 \) and \( 2q - p + 2 \). Since the triangle is equilateral, these sides must be equal:

\[
2p - q + 5 = 2q - p + 2
\]

### Step 1: Solve for \( p \) and \( q \)

Rearranging the equation:

\[
3p = 3q - 3 \implies p = q - 1
\]

### Step 2: Substitute \( p = q - 1 \) into one side

Substituting into the expression for the side:

\[
2(q - 1) - q + 5 = q + 3
\]

So the side length is \( q + 3 \).

### Step 3: Calculate \( k \)

Given that the area of the triangle is \( \frac{\sqrt{3}}{4} \times k \), and using the formula for the area of an equilateral triangle \( \frac{\sqrt{3}}{4} \times \text{side}^2 \), we have:

\[
k = (q + 3)^2
\]

Since \( k = (q + 3)^2 \), for the area to be expressed as \( \frac{\sqrt{3}}{4}k \), \( k \) must be the square of the side length \( q + 3 \).

### Result:
The value of \( k \) depends on the specific value of \( q \). However, if no specific value for \( q \) is provided, the answer will be \( (q + 3)^2 \).

If \( q = 0 \) (which is the simplest assumption for this type of problem):

\[
k = (0 + 3)^2 = 9
\]

Thus, the numerical value of \( k \) is **9**.

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### Would you like more details, or do you have any further questions?

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**Related Questions:**

1. What would \( k \) be if \( q = 1 \)?
2. How does the side length change if \( p \) is doubled?
3. What is the perimeter of the triangle when \( k = 16 \)?
4. How can we calculate the height of the equilateral triangle using \( k \)?
5. How does \( k \) change if \( p = q \)?

**Tip:** Always ensure the expressions are fully simplified before substituting values.

Learn how to calculate the area of an equilateral triangle with sides defined by expressions 2p - q + 5 and 2q - p + 2. This problem involves solving for the side length, determining the area formula, and finding the numerical value of k based on specific values of p and q. Suitable for students in Grades 10-12.

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