In this problem, we have a triangle $PQR$ in the Cartesian plane with vertices at points $P(-4, 2)$, $Q(5, 2)$, and $R(3, a)$. Given that $PQ$ serves as the base of the triangle and the height of the triangle is $5$ units, we need to determine the correct value(s) of $a$ based on the provided statements.

Solution Steps:

1. Determine the Length of $PQ$: Since points $P$ and $Q$ share the same $y$-coordinate (2), $PQ$ is a horizontal line segment. The length $PQ$ can be calculated as: $PQ = |x_Q - x_P| = |5 - (-4)| = 9$

2. Height Condition: The height of the triangle from point $R$ to the line $PQ$ is given as $5$ units. Since $PQ$ lies on the line $y = 2$, the vertical distance (height) from $R(3, a)$ to this line is $|a - 2|$.

   Therefore, we set up the equation: $|a - 2| = 5$

3. Solving for $a$: Solving the absolute value equation $|a - 2| = 5$, we get: $a - 2 = 5 \Rightarrow a = 7$ $a - 2 = -5 \Rightarrow a = -3$

   Thus, the possible values of $a$ are $7$ and $-3$.

4. Verification of Statements:

   • Statement (1): $a = 7$ – This is true, as we found $a = 7$ is one solution.
   • Statement (2): $a = 2$ – This is incorrect since $a = 2$ would make the height zero, not 5.
   • Statement (3): $a = -3$ – This is true, as we found $a = -3$ is the other solution.
   • Statement (4): $a$ cannot be determined – This is incorrect, as we have determined possible values for $a$ (7 and -3).

5. Conclusion: Based on the correct statements, only statements (1) and (3) are true.

Answer:

The correct answer is Option B: (1) and (3) SAJA yang benar.

Would you like further clarification on any part of this solution?

1. What is the method to calculate the length between two points in a coordinate plane?
2. How does the absolute value function affect the calculation in this context?
3. Why is the height measured as an absolute difference in this problem?
4. Could there be scenarios where $a = 2$ might be correct in a different setup?
5. What are some other cases where triangle properties are used to solve for unknowns in geometry?

Tip: When working with absolute values, remember that they represent distances, which are always non-negative, resulting in two possible cases for solutions.