Answer :
To find the distance between the two points (3, 10) and (9, 4), we use the distance formula which is based on the Pythagorean theorem. This formula states that the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a 2-dimensional Cartesian coordinate system is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Given our points (3, 10) and (9, 4), let's denote them as \((x_1, y_1) = (3, 10)\) and \((x_2, y_2) = (9, 4)\). Substituting these values into the distance formula, we get:
\[
\begin{align*}
d &= \sqrt{(9 - 3)^2 + (4 - 10)^2} \\
d &= \sqrt{6^2 + (-6)^2} \\
d &= \sqrt{36 + 36} \\
d &= \sqrt{72} \\
\end{align*}
\]
Now, simplify the square root. We can see that 72 can be broken down into \( 36 \times 2 \), and we know that the square root of 36 is 6. Therefore:
\[ d = \sqrt{36 \times 2} = 6\sqrt{2} \]
The exact value of the distance is \( 6\sqrt{2} \), but if we want to round it to the nearest tenth, we need to approximate the value of \( \sqrt{2} \), which is approximately 1.414. So:
\[ d \approx 6 \times 1.414 \]
\[ d \approx 8.484 \]
Rounded to the nearest tenth, the distance \( d \) is approximately 8.5 units.
