Coordinate Plane Geometry
Distance between two points– plot the two points, create a right triangle (horizontal and vertical legs- the “x-leg” and “y-leg), find the lengths of the legs, use the pythagorean theorem to find the hypotenuse.
Midpoint– plot the two points, create a right triangle (horizontal and vertical legs- the “x-leg” and “y-leg), find the lengths of the legs, find half of those lengths, add or subtract those new distances to the x and y values accordingly (all depends on if you are going up, down, left or right). Don’t forget to be careful not to mix up your x and y values.
Quick way for midpoint of two points: Find the midpoint for the x‘s, find the midpoint for the y‘s. Those give you the coordinates of your midpoint. Remember, the shortcut for midpoint is to just find the average (add and divide by two). Be careful about negatives and how they work when you add or subtract them!
Equations of Lines
y=mx+b (m is the slope, b is the y-intercept)
You don’t need to memorize which is the slope and which is the y-intercept to plot points to create a line! Just plug in a number for x, follow the “rule” (multiply by “m”, add “b”), and record your output. That pair of x and y is an (x, y) coordinate that “works” with your equation and is therefore on the graph of the line. All of the points on the graph of a line are solutions to the equation of the line.
All you need to graph a line is any two example points!
Parallel lines have the same slope!
Slope is rise over run: (y2-y1 over x2-x1) You can remember that the y’s go on top because we say rise over run. Rise has to do with up and down which has to do with y-values.
Perpendicular lines have slopes that are opposite reciprocals. (Opposite of positive is negative, opposite of negative is positive, reciprocal means to flip: for integers like 3 and -7, their reciprocals are 1/3 and -1/7 respectively.) Opposite reciprocal means to do both!
Remember also that perpendicular means to meet at right angles.
To classify quads on the coordinate plane given 4 vertices, it can help to plot the points to get an idea of what the shape looks like and create a conjecture (this is not necessary though). To officially determine or prove the classification of a shape though, you need to do things like find slopes, lengths, and midpoints to prove properties like parallel lines, congruent segments, or bisectors are actually occurring.
Area is how many square units can fit into a figure.
For rectangles, you just multiply the two side lengths.
For triangles, we think of them as half of a rectangle, so we do the base times the height, divided by 2.
Important: The base and the height of a triangle must be perpendicular!
The height of a triangle does not need to be a side of the triangle, or even inside the triangle.
For parallelograms, similar to rectangles, we do base times height, but once again like triangles, the height and the base must be perpendicular! The height of a parallelogram is not one of the sides of the parallelogram.