It looks like you've provided two points: (1,2) and (0,-4). Here are a few mathematical tasks we can perform with these points:
1. **Find the distance between the points:**
The formula to find the distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Plugging in the coordinates:
\[
d = \sqrt{(0 - 1)^2 + (-4 - 2)^2}
= \sqrt{(-1)^2 + (-6)^2}
= \sqrt{1 + 36}
= \sqrt{37}
\]
So, the distance between the points is \(\sqrt{37}\).
2. **Find the midpoint of the line segment connecting the points:**
The formula for the midpoint \((x_m, y_m)\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
\left( x_m, y_m \right) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Plugging in the coordinates:
\[
\left( x_m, y_m \right) = \left( \frac{1 + 0}{2}, \frac{2 - 4}{2} \right)
= \left( \frac{1}{2}, -1 \right)
\]
So, the midpoint is \(\left( \frac{1}{2}, -1 \right)\).
3. **Find the slope of the line passing through the points:**
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in the coordinates:
\[
m = \frac{-4 - 2}{0 - 1}
= \frac{-6}{-1}
= 6
\]
So, the slope of the line is \(6\).
4. **Find the equation of the line passing through the points:**
Using the point-slope form of the equation of a line \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is one of the points:
We already have the slope \(m = 6\), and we can use point \((1, 2)\):
\[
y - 2 = 6(x - 1)
\]
Simplifying:
\[
y - 2 = 6x - 6
\]
\[
y = 6x - 4
\]
So, the equation of the line is \(y = 6x - 4\).
Let me know if there is anything specific you would like to explore with these points!
(1,2) and (0,-4)
1 answer