Question

Lulu marks the points (9,1) and (-3,6) on a coordinate plane. What is the distance between these two points, rounded to the nearest tenth of a unit? What is the midpoint of the segment? If not a whole number, write as a decimal.

Answer 1

To find the distance between the points \( (9,1) \) and \( (-3,6) \), we can use the distance formula:

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

Here, \( (x_1, y_1) = (9, 1) \) and \( (x_2, y_2) = (-3, 6) \).

First, we find the differences:

\[
x_2 - x_1 = -3 - 9 = -12
\]
\[
y_2 - y_1 = 6 - 1 = 5
\]

Next, we can substitute these values into the distance formula:

\[
d = \sqrt{(-12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13
\]

Next, to find the midpoint of the segment connecting the two points, we use the midpoint formula:

\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Substituting the coordinates of the points:

\[
\text{Midpoint} = \left( \frac{9 + (-3)}{2}, \frac{1 + 6}{2} \right) = \left( \frac{9 - 3}{2}, \frac{7}{2} \right) = \left( \frac{6}{2}, \frac{7}{2} \right) = (3, 3.5)
\]

In summary:

- The distance between the points rounded to the nearest tenth is \( 13.0 \).
- The midpoint of the segment is \( (3, 3.5) \).