003 FOM - Geometry Flashcards
What are vertical angles? What is true about them?
Opposite angles created when two straight lines intersect. They have the same measure.
Define a right angle.
An angle with the measure of 90 degrees.
How is a perpendicular line denoted?
By an upside down T.
Define a polygon.
A closed plane figure formed by three or more line segments (each a “side” of the polygon).
Define vertices.
The points of intersection on a polygon.
Define convex polygon.
Polygon where each interior angle has a measure of less than 180 degrees.
Define a quadrilateral
Polygon with four sides
Define a pentagon.
Polygon with five sides.
Define hexagon
Polygon with six sides.
Express the sum of the interior angles of a polygon as an equation.
180 degrees x (n-2)
Where n is the number of sides of the polygon.
Define isosceles triangle.
At least two sides of the triangle are of equal length.
What is true about sides/angles in an isosceles triangle?
If two sides are the same length, their opposite angles are the same length (and vice versa).
What are the components of a right triangle?
Sides forming right angle are LEGS. Third side opposite the right angle (and the longest side) is the HYPOTENUSE.
Define Pythagorean theorem.
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
Define right triangle
A triangle that has a right angle. Also any triangle in which the lengths of the sides are in the ratio 3:4:5.
What is the altitude of a triangle?
The segment drawn from a vertex perpendicular to the side opposite that vertex.
What is the base of a triangle?
Relative to a vertex and an altitude, the opposite side (ie the side where the altitude connects to). Any side of a triangle can serve as a base (whatever side the height is perpendicular to).
What is the area of a triangle?
(length of altitude * length of the base) / 2
aka, the average of the altitude and base
OR 1/2(base*height)
What is the ratio of sides in a 45-45-90 degree triangle?
1:1: sq root 2
In triangle JKL with hypotenuse KL, if JL =2 and JK = 2, then KL = 2sq rt 2
What is the ratio of sides in a 30-60-90 triangle?
1: sq rt 3: 2
For triangle XYZ with hypotenuse YZ, if XZ = 3, then XY = 3sqrt3 and YZ = 2X3=6
Define a line segment
A part of a line (i.e., a straight line that extends without end in both directions) between two endpoints on the line.
Define quadrilateral
Polygon with four sides
Define parallelogram
Quadrilateral in which both pairs of opposite sides are parallel and equal.
Define rectangle vs. square
Parallelogram with right angles. Square is a rectangle with all sides of equal length.
Define trapezoid
Quadrilateral with two sides parallel. Third and fourth sides are slightly diagonal.
What is the area of a trapezoid?
(1/2)(sum of the lengths of the bases)(the height)
Define a circle
Set of points in a plane that are located the same distance from a fixed point (the center of the circle).
Define chord in the context of a circle.
Line segment that has its endpoints on the circle.
Define diameter.
A chord that passes through the center of the circle.
Define radius
Segment from the center of the circle to a point on the circle. Equals one-half of the diameter.
Define circumference of a circle.
The perimeter of the circle. Defined as 2pir, where pi is approximately 3.14, and r is the radius.
Define area of a circle
pi*r^2 where pi is about 3.14 and r is the radius
How to find the length of an arc in a circle?
(x/360) where x is the degrees of the angle formed by the two radii that form the boundaries of the arc.
What is a tangent line in relation to a circle?
A line with one point in common with a circle (ie the point of tangency).
Define inscribed.
Being contained within something else.
Define circumscribed
Containing something else.
Relate right triangles to circles.
One of the sides of a right triangle that is inscribed in a circle will be the diameter of the circle.
What is a rectangular solid?
A 3D figure formed by 6 rectangular surfaces, each a face. Has 6 faces, 12 edges, and 8 vertices.
What is the surface area of a rectangular solid?
Sum of all the areas of the faces.
What is the volume of a rectangular solid?
Length x width x height
aka: area of the base x height
What is the surface area of a cylinder?
2(pir^2) + 2pirh
aka the sum of the areas of the two bases, plus the area of the curved surface
What is the area of a cylinder?
pir^2h
aka the area of the base x the height
What is the x axis?
The horizontal line on a coordinate plane
What is the origin?
Where the x and y axes intersect on a coordinate plane.
What are the divisions of a plane called?
Quadrants
Express a point on the coordinate plane
(x,y)
How to find the distance between two points on a plane?
Pythagorean theorum using a right triangle where the two points are two endpoints of a hypothetical triangle, with the straight line between the two serving as the hypotenuse of the triangle.
What is the equation of a line on a plane? A vertical line? Define the terms
y = mx + b where m is the slope, and constant term b is the y-intercept
vertical line: x=a
What is the slope of a line?
(difference in y coordinates)/(difference in x coordinates)
i.e., rise/run
How do you find the x intercept of a line?
Set y = 0 and solve for x in the equation y=mx+b
What is true about a circle?
If you know one thing about it (radius, diameter, circumference, area), you can determine everything else.
d = 2r; c = 2pir (aka dpi); a = pi*r^2
Define the relationship between circumference and diameter.
c = d*pi, therefore c/d = pi
Define sector
A fractional portion of the circle (slice of pizza).
Define arc length
The portion of the circle’s arc that is bound by a sector.
Define central angle.
A measure of the degrees between two radii, which make up the outer bounds of a circle’s sector.
What is the formula for area of a sector?
Area of the circle * (central angle/360)
What is the formula for arc length of a sector?
2pi*r x (central angle/360)
What is true about the sides of a triangle a, b and c?
1) The sum of a + b > c.
2) The difference of a-b
What is true about the interior angles of a triangle?
They add up to 180 degrees
What is true about the GMAT and geometrical shapes?
They are not always drawn to scale!
What is the relationship between a triangle’s interior angles and the length of its sides?
The longest side is opposite the largest angle.
What is true about the height of a triangle?
It can be outside the triangle - just extend the base.
Define hypotenuse.
The side of a triangle opposite the right angle.
What are the sides of a right triangle besides the hypotenuse called?
The legs.
Define Pythagorean triplet.
A right triangle in which all three sides have lengths that are integers.
Name the 5 most common Pythagorean triplets.
3-4-5 (and doubled, 6-8-10)
5-12-13 (and doubled, 10-24-26)
8-15-17
What is true about Pythagorean triplets?
You can scale them in size by applying a common multiplier.
What is true about splitting quadrilateral shapes?
You can always split them into two triangles.
What do hash marks in a shape signify?
Equal lengths or angles.
What do arrows in shapes signify?
Parallel lines
How do you find the area of a parallelogram?
Base x height
What is true about the interior angles of a parallelogram?
Opposite angles are also equal, and adjacent angles add up to 180 degrees.
What’s the difference between a line and a plane?
A line is one-dimensional, because you only need one number to identify a point’s location. A plane is two-dimensional because you need two numbers to identify a point’s location.
What should you do if you know just one coordinate?
Narrow down the possibilities by drawing a line
What are the quadrants of a coordinate plane.
1-4 going clockwise, with 1 in the upper right-hand corner.
What does y = 2x+4 indicate?
The line crosses the y-intercept at +4. The line slopes up and to the right, with two increases in y (rise) for every increase in x (run)
