Be careful of geometry problems where the figure is drawn to LOOK parallel even though it isn't.
Only go by what you are told. Don't assume away!!
Redrawing figures is a way to avoid getting mixed up from the problem.
GMAT will try to trick you by drawing figures that are not to scale, aka they will lie about the figure so that I make incorrect assumptions!!
If I redraw the figure based on what I am told about the figure (e.g. these lines are parallel but these are not), then I can avoid these problems and not get tripped up on bad assumptions.
Quadrilaterals - advanced
A common GMAT problem is to MAXIMIZE the area of a quadrilateral, given a fixed perimeter.
SQUARES will always have the largest area of any quadrilateral with fixed perimeter!
--> e.g. if perimeter of a quadrilateral is 25, then the side length for the quadrilateral with biggest area is 6.25
Polygons - advanced
A regular polgon with all sides equals will maximize area for a given perimeter, and minimize perimeter for a given area
Maximum area of parallelogram or triangle - advanced
If you are given two sides of a triangle or parallelogram, you can maximize the area by placing those 2 sides perpendicular to each other.
To rationalize this rule of thumb, think about the area formulas for triangles (0.5*base*height) and parallelograms (base*height).
The base will be constant while the height changes. Therefore, the height needs to be maximized (since the base will be constant) to get the largest area. Otherwise, you keep getting something smaller, which will involve an angle less than 90 degrees.
Parabolas - advanced
To find where a parabola touches the x-axis, I can plug in y=0 into the quadratic formula to find the solutions for x.
x = [-b ± sqrt(b^2 - 4ac)] / 2a
To quickly know how many solutions the equation has, just look at [b^2 - 4ac].
(1) if >0, then the parabola crosses x-axis twice
(2) if =0, then the parabola touches x-axis once
(3) if <0, then the parabola never touches the x-axis
Perpendicular bisectors - advanced
+ The perpendicular bisector of a line segment has the negative reciprocal slope of the line segment that it's bisecting
+ The product of the 2 slopes of 2 lines that are perpendicular is -1.
+ Exception is when one line is horizontal and one is vertical
Intersecting vs. parallel lines - advanced
With straight lines, there are three possibilities when dealing with these lines and the possibility of them intersecting.
It's especially important to keep these 3 possibilities in mind when dealing with DS questions.
(1) that two lines intersect once and there is one (x,y) that satisfies the equations for both lines
(2) that both lines are parallel and that there are no solutions (x,y) that satisfies the equations for both lines
(3) that both lines are the same and that there are infinitely many solutions (x,y) that satisfies both lines