# Week 11 – Math 10

This week we spent most of the time readying for our unit test and taking the test. Plus, this week was only three days long, one of which with an early dismissal. In this short amount of time however, we did learn something new. How to find the distances between pints and the midpoint of a line.

Sometimes you can count the space between two points if they are close together, but sometimes they are so far apart, that it would be ineffective to count. To find the distance between two points this far apart, you need their coordinates. If this line is completely horizontal or completely vertical, you can find the distance with just the x or y coordinate. If a line is horizontal, you just need the x coordinate, and if it’s vertical, just the y.

ex.

Let’s say we have a horizontal line with the coordinates (10,12) and (24,12). To find the length, you subtract one x point from the other. But what’s important to remember is that the distance between the points will always be positive, so we can do 24-10=14, which is positive. But we can also do 10-24=-14, all you have to do is take away the negative.

To find the midpoint, it is pretty simple. If you have the distance between the points, you can just take it and divide it in two. The midpoint is the exact point in the middle of a line. Similarly to eyeballing distance, you can sometimes find the midpoint without trying too hard, but sometimes, the midpoint is not on a line, and is somewhere in the middle, where eyeballing is more difficult. The best way to find it is to use the math.

ex.

With the same example, the midpoint would be somewhere between 10 and 24. The distance was 14, so the midpoint would be half of that. 14÷2=7. The midpoint is 7.

If the line we are using is an oblique line (meaning its on an angle), we must make a triangle. Wherever the lines two points make touch, marks the right angle point of the triangle.

From here, you need to use pythagorean theorem ( $a^2 + b^2 = c^2$). Since the adjacent and opposite sides are both straight, you can find their distance. Just use the coordinates of where the two lines meet. Use pythagorean theorem to find the distance of the original line. Sometimes the square root of $c^2$ in the equation is a number followed by multiple points. To find the exact answer, you can just leave it as the square root of ___.

To find the midpoint, you do something similar to what we did with straight parallel lines, but because an oblique line is not parallel to any axis, you need to use both the x and y axis to find coordinates, which in turn give you the midpoint.

ex.

Let’s say we have a line with the points (2,1) and (5,4). The distance of this line is the square root of 18. To find the midpoint, we use 2 (from (2,1)), and 5 (from (5,4)), 5+2=7, 7÷2=3.5. Now we do the same with the other two: 1+4=5, 5÷2=2.5, therefor the coordinates of the midpoint of the line is (3.5,2.5), which is in-between lines on both the x and y axis.