Week 6, Day 1: Geometry Review
Chapter 15: Math Foundations – Geometry Review
So many formulas! Remember: memorization works best with
repetition. Make some flash cards! On paper! You can get bonus points in my
class this way!
Lines and Angles
Lines are geometric abstractions, only represented by
the image; they are actually infinitely long and have no width. A straight line
is the shortest distance between two points. They are named by those points.
Line segments are finite portions of lines,
also named by two points.
Parallel lines never intersect and are indicated with
two pipes: ||
A vertex (pl. vertices) is where two lines intersect
to form an angle. An angle can be named by the angle itself, or by a
combination of the three points that form the lines and the vertex.
Sum of angles around a point
The sum of angles around a single point is 360
degrees.
Sum of angles along a straight line
Along a straight line, in contrast, the sum of the angles is
one half of 360 or 180 degrees. When two angles add up to 180 degrees, they are
called supplementary.
Perpendicularity and right angles
On the other hand, when two angles add up to be 90 degrees,
they are called complementary. A 90 degree angle is a right angle.
When two lines intersect at a right angle, they are called perpendicular,
and the symbol is an upside down T: ^.
Angle bisectors
Anything bisected is divided in half; an angle bisector
is split into two equal angles, while a line bisector is divided into two equal
lengths.
Adjacent and vertical angles
Adjacent angles share a side, but vertical angles
are opposite each other. Vertical angles are also called opposite angles.
Adjacent angles are always supplementary, and opposite angles are always equal
in measure.
Angles around parallel lines intersected by a transversal
Because of the properties of angles and lines, when two
parallel lines are intersected by a third line, called a transversal, we
can calculate the measure of any angle in the diagram based on the measure of
only a single given angle.
All the acute (smaller than 90 degrees) angles will
be exactly equal to each other, all the obtuse (larger than 90 degrees
but less than 180 degrees) angles will be exactly equal to each other, and the
acute angles will be supplementary to the obtuse angles.
Polygons
Polygons are all closed figures with straight line segments
for edges. We name them according to the number of sides and angles they have,
usually with Greek prefixes. Triangles have three sides, quadrilaterals
have four, pentagons have five, etc.
Perimeters are the sum of the sides of the polygon.
The vertex of a polygon is just one of the angles, usually
labeled with a letter. The polygon is then named by the vertices.
A diagonal in a polygon is an imaginary line
connecting any two vertices that are not connected by a side.
A regular polygon has exactly equal side lengths
and angle measurements.
The sum of the interior angles of a polygon is always
based on the number of angles. 180 multiplied by the number of angles minus 2
will give you the sum of the interior angles.
Triangles
Important terms
Triangles are three-sided figures with three interior
angles. A right triangle has one 90 degree angle. The sum of
angles in a triangle is always 180 degrees. The hypotenuse is always
the longest side of the triangle, opposite the 90 degree angle. An isosceles
triangle has two equal sides and angles, but an equilateral triangle
has three equal sides and angles, each angle being exactly 60 degrees. Lastly,
note that on an isosceles triangle, the legs are the two equal sides,
but on a right triangle, the legs are the two sides which are adjacent to the
right angle. The altitude or height of the triangle is how tall the
triangle is; usually we can use one of the legs to measure the height.
Sides and angles
Note that the sum of any two sides will ALWAYS be greater
than the length of a third side on a triangle.
Perimeter and area of triangles
There is no special formula for the perimeter of a triangle,
but the area of a triangle is ½ the base times the height.
Right triangles
The hypotenuse is always opposite the right angle, which is
always the largest angle in a right triangle. We use the Pythagorean theorem
to calculate the side lengths, were the sum of the squares of the legs equals
the square of the hypotenuse, or a2 + b2 = c2.
Pythagorean triples
However, there are patterns in the Pythagorean theorem which
we call Pythagorean triples. These include the most common 3, 4, 5
triangle where one leg is three units, another is four, and the hypotenuse is
five. Other triples can be multiples of the 3, 4, 5, or one of the other
triples listed on your memorization chart, such as 5, 12, 13.
Special right triangles
There are two special right triangles: 45-45-90, which has
side lengths of x, x, and xÖ2, and the 30-60-90, which has side lengths of x, xÖ3,
and 2x. This is the ratio between the sides.
If a right triangle has one 45 degree angle, the other angle
has to be 45 degrees as well. This means that you have an isosceles right
triangle, so the measure of one leg will equal the other leg, and the
hypotenuse will be that leg length times the square root of two.
If a right triangle has one 30 degree angle, the other angle
has to be 60 degrees. The shortest side will then be multiplied by 2 to get the
hypotenuse, and the other leg will be that shortest side length multiplied by
the square root of three. The shortest side will be opposite the smallest, 30
degree angle.
Triangles and quantitative comparison
Given the properties of triangles: if you know two angles,
you can calculate the third (because all triangle angles will sum to equal 180
degrees); to find the area, you MUST have both the base and the height of the
triangle; in a right triangle, you can find the third side if you have two
sides, and having two sides means you can find the area as well; and in both of
the special right triangles, if you have one side, you can find everything else.
HOWEVER, be careful about making assumptions – if they do
not tell you explicitly that a side length is the hypotenuse/longest side, do
not assume that it is, for example.
Quadrilaterals
Any four-sided polygon is a quadrilateral. This includes squares,
rectangles, parallelograms, rhombuses, and trapezoids.
Squares have all equal sides; rectangles have two pairs of equal sides. A
parallelogram has two pairs of parallel sides, but has no 90 degree angles. A
trapezoid has only one pair of parallel sides, and the legs will only be equal
in an isosceles trapezoid.
Perimeters of quadrilaterals
You simply add the sides, once again. But you can also say twice
the length plus twice the width, or twice the base plus twice the height for
rectangles; for squares you just say four times the side measure.
Areas of quadrilaterals
These are all some variety of base times height or length
times width. A square is simply one side length squared. A rectangle is base
times height or length times width, and so is a parallelogram. A trapezoid is
special: ½ times the height times the sum of the parallel sides.
Lastly, note that in a parallelogram or rectangle, knowing
two adjacent sides gives you all the side measures; in a parallelogram two
adjacent angles will give you all the other angles, too. A rectangle’s area can
be calculated just with two adjacent sides. In a square, knowing one side
length gives you all the information you need for area, perimeter, etc.
Circles
Important terms
Technically, a circle is all points equidistant from the
center point, usually named by a letter, which is also the name of the circle.
The diameter is the line from edge to edge that passes through the
center point, and is twice the length of the radius; a radius, in other
words, goes from the center to the edge, and is one half the diameter. A central
angle is formed by two radii (plural of radius). A chord is a line
that passes from one edge to one edge but not necessarily through the center;
the diameter is the longest possible chord. Lastly, the tangent is a
line outside the circle that touches the edge in one point and is perpendicular
to a radius.
Circumference and arc length
The circumference is the perimeter of the circle. It is
always equal to pi times diameter, or pi times twice the radius. Arc length is
measured as part of the circumference, or as an equal measure to the central
angle.
Area and sector area formulas
Area is pi times the radius squared. A sector is a portion
of the area; if you can calculate the central angle, you can divide the area by
the portion that the sector represents.
Coordinate Geometry
Important terms and concepts
Plane – (particularly the Cartesian or coordinate plane) – a
flat surface extending infinitely in all directions; the Cartesian/coordinate
plane has two axes, x and y
x-axis and y-axis – (note: axis (sg), axes (pl)) – horizontal
(x) and vertical (y) number lines intersecting perpendicularly to indicate
location on the coordinate plane
ordered pair – values on the x- and y-axes indicating
horizontal and vertical location
coordinates – the numbers designating distance from the axes
origin: (0,0), the point at which the axes intersect
Plotting points
Every point on the coordinate plane has an ordered pair that
tells you its location. When the point lies on the x-axis, the y value is zero,
and when the point lies on the y-axis, the x value is zero. Starting at the
origin, moving up and to the right, both values are positive; to up and to the
left, the x value is negative and the y value is positive; down and to the
left, both values are negative; down and to the right, the x value is positive,
and the y value is negative. In other words, by quadrant, counter-clockwise, I
is (+, +), II is (-, +), III is (-, -), and IV is (+, -).
Distances on the coordinate plane
The distance between any two points can be measured as a
straight-line segment between those points; when the line segment is parallel
to the x-axis, the y values are all identical, and when the line segment is
parallel to the y-axis, the x values are all identical. If the line segment is
parallel to the axes, you can simply subtract one end point’s value from the
other. If the line segment is not parallel, you can treat it as the hypotenuse
of a triangle and find its length using the Pythagorean theorem, or use the
distance formula from our memorization sheet.
Equations of lines
Straight lines can be described by linear equations, usually
a form of y = mx + b, where m = slope and b = y-intercept (where the line
crosses the y-axis). Slope is calculated as the change in y divided by the
change in x, or rise over run. When the slope is equal to zero, y = b and the line
will be parallel to the x-axis. Lines that are parallel to the y-axis will have
x = a, where a is the x-intercept (where the line crosses the x-axis).
Graphing functions and circles
A function is for our purposes just another way to write the
equation of a line, where f(x) = y. This is a kind of notation that we have
seen used elsewhere for false operations with strange symbols.
Graphing quadratic functions
Quadratic equations produce parabolas instead of straight
lines. The simplest example as given in our book is y = x2. In f(x)
= ax2 + bx + c, a determines which direction the parabola opens; if
positive, the parabola opens upward, and if negative, the parabola opens
downward. The c value represents vertical shift: a negative value represents how
far below the x-axis the parabola moves, while a positive value represents how
far above the x-axis the parabola moves. The left-right shift is less
immediately obvious, but is based on adding or subtracting values from x (see
p. 327). Multiplying values over the whole function will expand or contract the
function, making it more broad or more narrow, while multiplying by a negative
number will flip the parabola’s direction along the x-axis.
Graphing circles
These are the most rare form of geometry question on the GRE,
but generally, the form of the equation for a circle is some variation of: (x –
a)2 + (y – b)2 = r2 where x and y are points
on the coordinate plane, and a, b, and r are constants. The radius, of course,
is r, and the coordinate of the center point of the circle are (a, b).
Solids
Important terms
Solid – three-dimensional figures, with dimensions of
length, width, and height (l, w, h) or height, width, and depth (h, w, d). Only
cubes and cylinders appear regularly on the GRE, but others are possible.
Uniform solid – solids that can be cut into parallel slices
of equal size and shape, aka congruent cross-sections.
Face – the surface of a solid lying in one plane.
Edge – line segment connecting adjacent faces.
Base – the bottom of a solid
Rectangular solid – a solid with six rectangular faces (a
brick).
Cube – special rectangular solids where all six faces are square.
Cylinder – a solid with circular top and base (a pipe or
can). A regular right cylinder has no slope to its long side, which is
perpendicular to the perfectly circular top and base.
Lateral surface of a cylinder – think of the paper you might
unwrap from a can of food – this can be measured as a rectangle, using the height
of the cylinder as the height of the rectangle, and the circumference of the
base as the length of the rectangle.
Formulas for volume and surface area
Volume of rectangular solids = area of base times the height
or length times width times height.
Surface area of rectangular solids = sum of areas of faces
or twice length times width plus twice length times height plus twice height
times width.
Cube volume = the length of one edge cubed.
Cube surface area = six times one edge squared.
Cylinder volume = pi times radius squared times height, or
the area of the circle times height.
Lateral surface area = twice pi times radius times height.
Cylinder surface area = twice pi times radius squared plus
twice pi times radius times height.
Multiple Figures
The key to these is always to divide the shape into smaller
shapes that you already know the formulas for – like triangles and rectangles.
Area of shaded regions
Always find the basic shapes that make up the different regions
of the larger shape. Find the area of the whole figure, then find the area(s)
of the unshaded region(s), and subtract the unshaded region(s) from the whole
to find the area of the shaded portion.
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