Question

what is the distance between (3, -4) and (8, 1)?
2√25
4√5
25√2
none of these

Explanation:

Step1: Recall Distance Formula

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Here, \((x_1, y_1) = (3, -4)\) and \((x_2, y_2) = (8, 1)\).

Step2: Substitute Values

Substitute into the formula: \( d = \sqrt{(8 - 3)^2 + (1 - (-4))^2} = \sqrt{(5)^2 + (5)^2} \).

Step3: Simplify Inside Radical

Calculate the squares: \( \sqrt{25 + 25} = \sqrt{50} \).

Step4: Simplify the Radical

Simplify \( \sqrt{50} \): \( \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \)? Wait, no, wait—wait, let's re - check. Wait, \((8 - 3)=5\), \((1 - (-4)) = 1 + 4 = 5\). So \((5)^2+(5)^2=25 + 25 = 50\). But the options given: Wait, the options are \(2\sqrt{25}\), \(4\sqrt{5}\), \(25\sqrt{2}\), None of these. Wait, \(2\sqrt{25}=2\times5 = 10\), \(4\sqrt{5}\approx4\times2.236 = 8.944\), \(25\sqrt{2}\approx35.355\). But our calculated distance is \(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\approx7.071\), which is not equal to any of the first three options. Wait, did I make a mistake? Wait, no, the points are \((3,-4)\) and \((8,1)\). Let's recalculate the distance formula:

\(d=\sqrt{(8 - 3)^2+(1-(-4))^2}=\sqrt{5^2 + 5^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}\). But the options are \(2\sqrt{25}\) (which is \(10\)), \(4\sqrt{5}\) (which is \(\sqrt{16\times5}=\sqrt{80}\)), \(25\sqrt{2}\) (which is \(\sqrt{625\times2}=\sqrt{1250}\)), and None of these. So the correct answer is "None of these".

Answer:

None of these