Find the distance between the points \( (4,-9) \) and \( (2,-1) \). Radical Form (simplest form): Decimal Form (nearest tenth): Decimal Form (nearest hundredth):

To find the distance between the points \( (4,-9) \) and \( (2,-1) \), we can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Replacing \( (x_1, y_1) \) with \( (4, -9) \) and \( (x_2, y_2) \) with \( (2, -1) \): \[ d = \sqrt{(2 - 4)^2 + (-1 - (-9))^2} \] Calculating further: 1. \( x_2 - x_1 = 2 - 4 = -2 \) 2. \( y_2 - y_1 = -1 - (-9) = -1 + 9 = 8 \) Now substituting back into the formula: \[ d = \sqrt{(-2)^2 + (8)^2} \] \[ d = \sqrt{4 + 64} = \sqrt{68} \] In simplest radical form, \( \sqrt{68} \) can be simplified: \[ \sqrt{68} = \sqrt{4 \cdot 17} = 2\sqrt{17} \] Now, for the decimal approximation: 1. Calculating \( 2\sqrt{17} \): \[ \sqrt{17} \approx 4.1231 \quad (\text{using a calculator}) \] Thus, \[ 2\sqrt{17} \approx 2 \cdot 4.1231 \approx 8.2462 \] So, rounding to the nearest tenth and hundredth: - Decimal Form (nearest tenth): \( 8.2 \) - Decimal Form (nearest hundredth): \( 8.25 \) In summary, the distances are: - Radical Form (simplest form): \( 2\sqrt{17} \) - Decimal Form (nearest tenth): \( 8.2 \) - Decimal Form (nearest hundredth): \( 8.25 \)