Questions: Persegi panjang ABCD memiliki luas 24 cm² dan keliling 20 cm. Jarak titik A ke titik C adalah ... cm.

Solution Steps

To find the distance between points A and C (the diagonal of the rectangle), we need to use the Pythagorean theorem. First, we need to determine the length and width of the rectangle using the given area and perimeter. Then, we can calculate the diagonal.

1. Use the area (length * width = 24) and perimeter (2 * (length + width) = 20) to form two equations.
2. Solve these equations to find the length and width.
3. Use the Pythagorean theorem to find the diagonal (distance from A to C).

We are given the area \( A \) and perimeter \( P \) of rectangle \( ABCD \): \[ A = \text{length} \times \text{width} = 24 \quad (1) \] \[ P = 2 \times (\text{length} + \text{width}) = 20 \quad (2) \]

From equation (2), we can simplify it to: \[ \text{length} + \text{width} = 10 \quad (3) \] Now we have a system of equations:

1. \( \text{length} \times \text{width} = 24 \)
2. \( \text{length} + \text{width} = 10 \)

Substituting \( \text{width} = 10 - \text{length} \) into equation (1): \[ \text{length} \times (10 - \text{length}) = 24 \] This simplifies to: \[ 10\text{length} - \text{length}^2 = 24 \] Rearranging gives us the quadratic equation: \[ \text{length}^2 - 10\text{length} + 24 = 0 \]

Using the quadratic formula \( \text{length} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ \text{length} = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} = \frac{10 \pm \sqrt{100 - 96}}{2} = \frac{10 \pm 2}{2} \] This gives us two possible solutions: \[ \text{length} = 6 \quad \text{and} \quad \text{width} = 4 \quad \text{or} \quad \text{length} = 4 \quad \text{and} \quad \text{width} = 6 \]

Using the Pythagorean theorem, the diagonal \( d \) is given by: \[ d = \sqrt{\text{length}^2 + \text{width}^2} \] Substituting the values: \[ d = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \approx 7.2111 \]

Final Answer