California High School Exit Exam Prep
Math Prep
Geometry and Measurement
Points, Lines and Planes
Finding the Measure of Segments
Ruler Postulate
The points of line segment can be paired with the real numbers such that one endpoint is zero and the other endpoint is a positive number.
Segment Addition Postulate
If Q is between P and R, then
PQ + QR = PR.
If PQ + QR = PR, then Q is between P and R.
Distance Formula
In
Geometry, the distance between two points is used to define the measure of a
segment.
Find distance between
(-2,3) and (5,-3)
Midpoint
and
are
Classifying Angles
Angle Bisector
An angle bisector cuts an angle into two congruent angles.
Perpendicular Lines
Perpendicular
lines intersect to form rights angles.
Angle AED = 45 degrees
Angle FEB = 90 degrees
Complementary Angles
Two angles are complementary if the sum of their measure equals 90 degrees.
The above angle equals a total 90 degrees. If the given angle equals 30 degrees, what does the complementary angle equal.
30 + what number = 90
30 + n = 90
30 + 60 = 90
So, the complement of 30 degrees is 60 degrees because 30 + 60 = 90.
Supplementary Angles
Two angles are supplementary if the sum of their measure equals 180 degrees.
The above angle equals a total of 180 degrees. If one of the angles equals 70 degrees what is its supplement?
70 + what number = 180
70 + n + 180
70 + 110 = 180
So, the supplement of 70 degrees is 110 degrees.
Parallel Lines
Parallel lines have the same slope OR lines are parallel if they can be extended forever and never cross.
Perpendicular Lines
Perpendicular Lines intersect to form four right angles.
Polygon
A polygon is a closed figure with three or more line segments that intersect at their endpoints.
Each line segment of a polygon is called a side.
The intersections at their endpoints are called vertices -- or singular vertex.
Polygons are named by the number of sides each has.
3 sided -- triangle
4 sided -- quadrilateral
5 sided -- pentagon
6 sided -- hexagon
7 sided -- heptagon
8 sided -- octagon
9 sided -- nonagon
10 sided -- decagon
The perimeter of a polygon is simply the measure around the sides.
THINK: If you wanted to know how far it was to walk around the outside of the mall you could measure around the outside of the mall. If you wanted to know how far it is around a polygon you just measure around the outside. So,
PERIMETER =
P = s + s + s + s
AREA =
Square: A = s x s or s2
Rectangle: A = lw
Parallelogram: A = bh
Triangle: A = 1/2 bh
Trapezoid: A = 1/2h(b1 + b2)
VOLUME =
Cube: V=(length of side)3
Rectangular Prism: V=length*width*height
Triangular Prism: V=1/2*length*width*height]
Cylinder: V=(area of the base)*height = śr2h
Sphere: V=volume=4/3śr3
Cone: V=1/3(Area of Base)(height) = 1/3(śr2)(height)
Pyramid: V=Area of the base * height * 1/3
Types of Triangles OR Triangle Classification
Triangles can be classified by their angles:
An
acute triangle has three acute angles.
An
obtuse triangle has one obtuse angle.
An
obtuse triangle has one right angle
Triangles can be classified by their sides:
Scalene: A scalene triangle has no equal sides.
Isosceles: An isosceles triangle has two sides of equal length and two angles of equal measure. The sides are called legs.
Equilateral: An equilateral triangle has three sides of equal length and three angles of equal measure. All angles are equal and all sides are equal.
The sum of the interior angles of a triangle are equal to 180o. To find the third angle of a triangle when the other two angles are known subtract the number of degrees in the other two angles from 180o.
Example:
How many degrees are in the third angle of a triangle whose other two angles are 35o and 70o?
180o - 35o -70o = 75o
The Pythagorean Theorem
In any right triangle, the square of the length of the hypotenuse* is equal to the sum of the squares of the lengths of the legs.
a2 + b2 = c2
42 + 62 = c2
* Remember that the hypotenuse is the side opposite the right angle. The other sides are the legs of the right triangle.
Measurement Conversion
The tests may ask you to compare weights, capacities, geometric measures, times and temperatures within and between measurement systems. Comparison can be easily done by the use of a simple proportion.
Example: Convert 3.4 miles to feet using a proportion. (1 mile = 5280 feet).
These measure must sometimes be expressed as rates.
Example: Thirty miles per hour is the same rate as which of the following?
a. 1 mile per minute
b. 2 miles per minute
c. 30 miles per minute
d. 30 minutes per mile
The answer is b. 2 miles per minute. The question asks us to find the equivalent rate to 30 miles per hour. There are 60 minutes in an hour. If one is traveling at half of that, or 30 miles per hour, then he/she is traveling 1 mile every 2 minutes or 30 miles per hour.
Circles
Circle Definitions
1. Radius: Line segment with endpoints on the circle and in the center of the circle. The radius is half of the diameter. A radius goes halfway across a circle.
2. Chord: Line segments with endpoints on the circle.
3. Diameter: Chord that goes through the center of a circle. d=2r
4. Tangent: A line or segment that intersects a circle at exactly one point.
5. Secant: A line or segment that intersects a circle at exactly two points.
6. Central Angle: An angle with a vertex is the center and sides are radii of the circle.
7. Arcs: Part of a circle.
8. Semicircle: An arc determined by the diameter.
xz is a diameter.
yw, yx and yz are radii.
Concentric circles have the same center but different radii.
AREA:
The area of a circle =
--
= approximately 3.14.
So, if the diameter of a given circle is 12, we know we can find the radius because it is half of the diameter.
So, if the radius = 6 then the area can be found using the formula:
So, the area of the given circle is 113.04
CIRCUMFERENCE:
The circumference of a circle is the distance around the outside of the circle. It could be called the perimeter of the circle.
The
circumference can be found by
So, if
and the diameter is 150, then
So, the circumference of the given circle is 471
