Question

find the distance between the points (1, 3) and (-2, 6). a 2 b 2√5 c 6 d 3√2

Explanation:

Step1: Recall distance formula

The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Step2: Identify coordinates

Here, \( x_1 = 1 \), \( y_1 = 3 \), \( x_2 = -2 \), \( y_2 = 6 \).

Step3: Substitute into formula

\( d = \sqrt{(-2 - 1)^2 + (6 - 3)^2} = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \)? Wait, no, wait: \( (-2 - 1) = -3 \), squared is 9; \( 6 - 3 = 3 \), squared is 9. Sum is 18? Wait, no, 9 + 9 is 18, square root of 18 is \( 3\sqrt{2} \)? But wait, let me check again. Wait, \( (-2 - 1) = -3 \), \( (6 - 3) = 3 \). So \( (-3)^2 + 3^2 = 9 + 9 = 18 \), square root of 18 is \( 3\sqrt{2} \), which is option D? Wait, but let me recalculate. Wait, maybe I made a mistake. Wait, \( x_2 - x_1 = -2 - 1 = -3 \), \( y_2 - y_1 = 6 - 3 = 3 \). Then squared terms: \( (-3)^2 = 9 \), \( 3^2 = 9 \). Sum is 18. Square root of 18 is \( \sqrt{9 \times 2} = 3\sqrt{2} \). So the answer should be D. Wait, but let me check the options again. Option D is \( 3\sqrt{2} \). So that's correct.

Wait, but wait, maybe I messed up the formula. Wait, distance formula is correct. So substituting the values:

\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

\( x_1 = 1 \), \( y_1 = 3 \); \( x_2 = -2 \), \( y_2 = 6 \)

So \( x_2 - x_1 = -2 - 1 = -3 \), \( y_2 - y_1 = 6 - 3 = 3 \)

Then \( (-3)^2 + 3^2 = 9 + 9 = 18 \)

\( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \), which is option D. So the correct answer is D.

Answer:

D. \( 3\sqrt{2} \)