Question

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Explanation:

Identify given coordinates and perimeter

Using the Coordinate Distance knowledge point
We are given three vertices of a quadrilateral (likely a rectangle):

$$ E(3, 5), \quad F(8, 5), \quad G(8, 1) $$

The total fencing available is \(22\) units. We need to find the fourth vertex \(H(x, y)\) such that the perimeter of the rectangle \(EFGH\) is at most \(22\) units.

Calculate the side lengths

Using the Coordinate Distance knowledge point

$$
LATEXBLOCK0
$$

Determine the fourth vertex

Using the Coordinate Distance knowledge point
For \(EFGH\) to form a rectangle, the opposite sides must be equal and parallel:

$$
LATEXBLOCK1
$$

Thus, the fourth vertex is \(H(3, 1)\).

Verify the perimeter constraint

Using the Coordinate Distance knowledge point

$$ \text{Perimeter} = 2 \times (EF + FG) = 2 \times (5 + 4) = 18 $$

Since \(18 \le 22\), the point \(H(3, 1)\) satisfies the fencing limit.

Answer:

• (A) \((3, 5)\)
• (B) \((-1, 1)\)
• (C) \((3, -3)\)
• (D) \((3, 1)\) (Correct answer)