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Geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of polynomials. When there is more than one variable, geometric considerations enter and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.
Math is something that every good student should know, but geometry is something that not everybody studies regularly. There are, however, several standardized tests that may require students to know a fair amount about geometry. There are also several tests that different institutions have created, such as the ACT and the SAT, which can test a student’s ability to think both quickly and efficiently. Colleges typically look at these scores to help them determine which students they want to have joining their student body. Because high school is such an important step on the way to college, it is very important to learn the things that will most help you, such as geometry.
Because there are so many standardized tests that require you to know a good amount of math, geometry included, it is important to know your subject. Not only is it important to know the geometry, however, it is also very important to do all you can in order to prepare for the test in other ways. Because the tests are usually timed, it is very important to learn how to prepare for tests before you go into them. Here are some tips that can get you started:
What Should You Know About Geometry?
When you go into a test, sometimes it is a bit harder than simply knowing the material that the teacher has said you will be tested on. When you go into standardized tests, you can’t always be sure what kinds of equations you will be asked to solve. Here are a few things that you might want to know:
Topics covered on the site:
Points are the simplest figures in geometry. A point has no size, although it may represent an object with size. It is shown pictorially as a dot and is usually named using a capital letter. All geometric figures consist of points.
A line is a set of points that originate from one point and extend indefinitely in two opposing directions. Often, a line is named by a lower case letter; if a line contains two points A and B, then the line can be denoted as AB or BA. Lines have no thickness, even though pictorial representations of lines do.
A plane can be modeled by a floor, a table top or a wall. Planes extend indefinitely in all directions and have no edges or thickness. Planes are often denoted by a single capital letter and represented as four-sided figures.
The Basic Building Blocks of Geometry
In geometry, the terms point, line and plane are considered undefined terms since they are explained using only examples and descriptions. Yet, these terms are useful in defining other geometric terms and properties.
Space is the set of all points.
Collinear points are points that lie in the same line.
Coplanar points are points that lie in the same plane.
Properties of Congruence
These are taken from the reflexive, symmetric and transitive properties of equality, from algebra.
Intersection of Geometric Figures
The intersection of two geometric figures is the set of points that are common to both figures.
1. intersection of a point and a line
2. intersection of two lines
3. intersection of a plane and a line
4. intersection of two planes
Properties of Parallel Lines:
If two parallel lines are intersected by a transversal, then the corresponding angles are congruent.
If two parallel lines are intersected by a transversal, then the alternate interior angles are congruent.
If two parallel lines are intersected by a transversal, then the same-side interior angles are supplementary.
If the transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other one as well.
If two lines are intersected by a transversal and their corresponding angles are congruent then, the lines are parallel.
If two lines are intersected by a transversal and their alternate interior angles are congruent then, the lines are parallel.
If two lines are intersected by a transversal and the same-side interior angles are supplementary then, the lines are parallel.
In a plane, if two lines are perpendicular to the same line then the lines are parallel.
Through a point not on the line, there is exactly one line parallel to the given line.
Through a point not on the line, there is exactly on line perpendicular to the given line.
If two lines are parallel to a third line then they are all parallell to each other.
If three parallel lines cut congruent segments off a transversal then, they cut off congruent segments on every transversal.
A line that contains the midpoint of one of the sides of a triangle and is parallel to another side bisects the third side of the triangle.
Parallel Lines:( || means parallel)
Parallel lines are lines that do not intersect and are coplanar.
Skew lines are lines that do not intersect and are not coplanar.
A line and a plane are parallel if they do not intersect.
If two parallel planes are cut by a third plane, then the lines of
intersection are parallel.
A line that intersects two or more coplanar lines in different points is called a transversal line.
Consider the figure:
Line t is the transversal of line m and n. The angles that are produced have special names.
Ã3,4,5,6 are called interior angles. Ã1,2,7,8 are called exterior angles.
The two non adjacent interior angles on opposite sides of the transversal are called the alternate interior angles.
Ã3 and Ã6 are alternate interior angles.
Ã4 and Ã5 are alternate interior angles.
The two interior angles on the same side of the transversal are called the same-side interior angles.
Ã3 and Ã5 are same-side interior angles.
Ã4 and Ã6 are same-side interior angles.
The two angles in corresponding positions relative to the two lines are called the corresponding angles.
Ã1 and Ã5 are corresponding angles.
Ã2 and Ã6 are corresponding angles.
Ã3 and Ã7 are corresponding angles.
Ã4 and Ã8 are corresponding angles.
In a plane, a figure has symmetry if there is an isometry, other than the identity, that maps the figure to itself.
For each figure there is a line of symmetry such that the reflection
maps the figure onto itself.
For each figure there is a point of symmetry such that a half turn
maps the figure onto itself.
Each figure has a center and rotates the figure onto itself.
Translation that maps the figure onto itself.
Segments and Rays
Consider the line:
Point B is between points A and C, and A, B and C are collinear.
Segment AC consists of points A and C and all points in between A and C. Points A and C are called the endpoints of the segment AC.Â
Segment AC, denoted as
Ray AC consist of points A and C and all points P such that C is between A and P. Point A is the endpoint of the ray. When denoting a ray, its endpoint is named first.
Ray AC, denoted as
Two rays with a common endpoint which extend indefinitely in opposing directions are called opposite rays. All the points on both rays are collinear with each other.
Postulate: The Ruler Postulate
The points on any line can be paired with the real numbers in such a way that:
Â Â 1. Any two chosen points can be paired with 0 and 1.
Â Â 2. The distance between any two points in a number line is the absolute. value of the difference of the real numbers corresponding to the points.
By virtue of the Ruler Postulate, a system to determine the length of a segment, which is equal to the distance between its endpoints, can be formulated. Every point on a number line can be paired to a real number; it is called the coordinate of the point. The length of a segment is found by determining the difference between the coordinates of its endpoints and taking the absolute value.
The coordinate of
A = -4.
B = -2
O = 0
C = 3
P = 5
The length of segment BC is determined as follows:
B has coordinate -2 , C has coordinate 3
-2 - ( 3 ) = -5
| -5 | = 5 \ the length of segment BC is 5.Â
Postulate: Segment Addition Postulate
Point B is a point on segment AC, i.e. B is between A and C, if and only if
AB + BC = AC
The Segment Addition Postulate is often used in geometric proofs to designate an arbitrary point on a segment. By choosing a point on the segment that has a certain relationship to other geometric figures, one can usually facilitate the completion of the proof in question.
Congruent segments are segments with equal lengths. In the above figure, AB and CD have the same length and are therefore congruent. This is denoted as AB = CD or AB @ CD
(" @ " means "is congruent to").
The midpoint of a segment is the point that divides the segment into two congruent segments. Since AB = CD, then C is the midpoint of segment AB.
The bisector of a segment is a line, plane, ray or another segment that intersects the segment at its midpoint. Since DC passes through C, the midpoint of AB, then DC is a bisector of segment AB.
he Midpoint Theorem: ( The Definition of Midpoint)
Given AB, its midpoint M is:
2AM = AB and AM = ½ AB
2MB = AB and MB = ½ AB
A perpendicular bisector of a segment is a line, ray or another segment that is perpendicular to the segment at its midpoint.
f a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.
f a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.
Angles are geometric figures formed by two rays having the same endpoint. The two rays are called the sides of the angle, and their common endpoint the vertex of the angle.
An angle is often denoted by the symbol Ã , followed by a number, a letter or three letters. If an angle is named using letters, the middle letter denotes the vertex of the angle, while the other letters refer to points on the sides of the angle; if there is only one letter, it then refers to the vertex. The above angle can be referred to as
Ã 1, Ã B or Ã ABC.
As the vertex B is the vertex of three different angles, the notation Ã B would be ambiguous. The three-letter rule for naming angles would be appropriate here. Thus, while Ã B is inexact in naming the above angles, ÃABC , ÃABD,
ÃCBD each refer to a unique angle.
Postulate: The Protractor Postulate
Given line AB and a point O on line AB. Consider rays OA and OB, as well as all the other rays that can be drawn, with O as an endpoint, on one side of line AB. These rays can be paired with the real numbers between 0 and 180 in such a way that:
Â Â 1. Ray OA is paired with 0, and ray OB is paired with 180.
Â Â 2. If ray OR is paired with a and ray OQ is paired with b,
Â Â Â Â Â then mÃ ROQ = | a - b |.
Angles can be classified by their measure:
Right angles are angles that measure 90°. It is often indicated by a square.
Acute angles are angles that measure between 0 and 90°.
Obtuse angles are angles that measure between 90 and 180°.
Straight angles are angles that measure 180°.
Postulate: Angle Addition Postulate
Point B lies in the interior of ? XYZ, if and only if, m?XYB + m?BYZ = m?XYZÂ
Like the Segment Addition Postulate, the Angle Addition Postulate is often used in geometric proofs when an arbitrary point is designated on the geometric figure in question, to facilitate the formulation of the required proof.
Complementary Angles are two angles whose measures sum up to 90Â°. The angles are complement of each other.
Supplementary Angles are two angles whose measures sum up to 180Â°. The angles are supplement of each other.
Congruent angles are angles that have equal measure.
Adjacent angles are two coplanar angles having a common vertex and a common side, but no common interior points.Â
Vertical angles are two angles whose sides form two pairs of opposite rays. When two lines intersect, two pairs of vertical angles are formed.
Vertical angles are congruent:
Perpendicular Lines:( ^ means perpendicular)
Perpendicular lines are two lines that form right angles.
Adjacent angles formed by perpendicular lines are congruent.
If two lines form congruent adjacent angles, then the lines are perpendicular.
If the exterior sides of two adjacent acute angles are perpendicular then the angles are complementary.
If two angles are supplements of congruent angles ( or of the same angle), then the two angles are congruent.
If two angles are complements of congruent angles ( or of the same angle), then the two angles are congruent.
A line contains at least two points, a plane contains at least three points but not all in one line, and space contains at least four points, but not all on one plane.
Through any two points, there is exactly one line.
Through any three points, there is at least one plane, and through any three noncollinear point there is exactly one plane.
If two points are in a plane then the line through the points are in that plane.
The intersection of two planes is a line.
The intersection of two lines is exactly at one point.
If line and a point not on the line exist, then a plane contains both
If two lines intersect, then a plane contains both of them.
The measured angles of a triangle sum up to 180°.
If two angles of a triangle are congruent to two other angles from another triangle then, the third angles are congruent.
The individual measured angles of an equiangular triangle is 60°.
In a triangle, at most, there can only be one right angle or obtuse angle.
In a triangle, the acute angles are complementary.
The measure of any exterior angle in a triangle is equal to the sum of the measures of the two remote interior angles.
Postulate: The SSS Postulate:
If three sides of a triangle is congruent with three sides of another triangle then, the triangles are congruent.
Postulate: The SAS Postulate:
If two sides and the included angle of a triangle are congruent with two sides and the included angle of another triangle then, the triangles are congruent.
Postulate: The ASA Postulate:
If two angles and the included side of a triangle are congruent with two angles and the included side of another triangle then, the triangles are congruent.
Theorem: The Isosceles Triangle Theorem:
If two sides of a triangle are congruent then the angles opposite those sides are congruent.
An equilateral triangle is also an equiangular triangle.
An equilateral triangle has three 60° angles.
The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.
Theorem: The AAS Theorem:
If two angles of a triangle are congruent, then the sides opposite the angles are congruent.
If two angles and the non-included side of a triangle are congruent to two angles and the non-included side of another triangle then , the triangles are congruent.
A median of a triangle is a segment from a vertex to the midpoint of the opposite side of the vertex.
An altitude of a triangle is a segment from a vertex and it is perpendicular to the opposite side of the vertex.
The segment whose endpoints are the midpoints of two sides of a triangle:
a) is parallel to the third side.
b) its length is half the length of the third side.
If one leg and the hypotenuse of a right triangle is congruent with
the hypotenuse and one leg of another right triangle then the right
triangles are congruent.
In right triangles, the side opposite the right angle is called the hypotenuse, and the other sides are called the legs.
The midpoint of the hypotenuse of a right triangle is equidistant from
the three vertices.
The Pythagorean Theorem
In a right triangle, the square of the hypotenuse is equal to the sum of
the squares of the legs.
c²= a²+ b²
If the square of one side of a triangle is equal to the sum of the
squares of the other sides then the triangle is a right triangle.
If the square of the longest side of a triangle is greater than the sum
of the squares of the other two sides then the triangle is an obtuse
If the square of the longest side of a triangle is less than the sum of
the squares of the other two sides then the triangle is an acute
Special Right Triangles
Isosceles Right Triangle:
The isosceles right triangle is also called the 45-45-90 triangle because of the measures of the angles.
Theorem: The 45-45-90 Triangle:
The hypotenuse of a 45-45-90 triangle is 2^½ times as long as a leg.
Theorem: The 30-60-90 Triangle:
The hypotenuse of a 30-60-90 triangle is twice as long as the short leg and the longer leg is 3^½ times longer than the shorter leg.
Polygons: (polygon means "many angles")
Polygons are made by coplanar segments such that:
1. Each segment exactly intersects two other segments, one at each
2. No two segments with a common endpoint are collinear.
convex polygon is a polygon that has no side in the interior of the polygon.
Polygons are named according to the number of sides they have. The triangle is simplest polygon. The terms that apply to triangles can also be applied to polygons.
A diagonal of a polygon is a segment that joins two non consecutive vertices.Â
The red dotted lines represent the diagonals of the polygon.
If the diagonals of a polygon is drawn from one vertex, then the sum of the measures of the angles of the polygon can be calculated.
The sum of the measured angles of an n sided polygon is (n-2)180.
number of sides = 5
(n - 2)180 = (5 - 2)180
(3)180 = 640Â°
The sum of the measures of the exterior angles of a convex polygon
is an angle at each vertex is 360.
A regular polygon is a polygon that is both equilateral and equiangular.
Assume temporarily that the conclusion is false and reason logically until a contradiction of the hypothesis or another fact is reached.
If one side of a triangle is longer than a second side, then the angle opposite the longer side is larger than the opposite angle of the second side.
If one angle of a triangle is larger than a second angle, then the side opposite the larger angle is longer than the opposite side of the second angle.
The perpendicular segment from a point to a line is the shortest segment from the point to the line.
The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.
Theorem: The Triangle Inequality:
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Theorem: The SAS Inequality Theorem:
If two sides of a triangle is congruent with two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second triangle then, the third side of the first triangle is longer than the third side of the second triangle.
Theorem: The SSS Inequality Theorem:
If two sides of a triangle is congruent with two sides of another triangle, but the third side of the first triangle is larger than the third side of the second triangle then, the included angle of the first triangle is larger than the included angle of the second triangle.
Ratios, Proportions and Similarities
The geometric mean between two numbers x and z is defined as
x/y = y/z
and y is called the geometric mean of x and z.
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to each other and to the original triangle.
When the altitude is drawn to the hypotenuse of the right triangle, the length of the altitude is the geometric mean between the new segments of the hypotenuse.
When the altitude is drawn to the hypotenuse of the right triangle, each leg is the geometric mean between the whole hypotenuse and the segment of the whole hypotenuse that is adjacent to that leg.
Postulate: The AA Similarity Postulate:
If two angles of a triangle are congruent with two angles of another triangle then the triangles are similar.
Theorem: The SAS Similarity Theorem:
If an angle of a triangle is congruent with another angle of a different triangle and their sides including the angles are in proportion then, the triangles are similar.
Theorem: The SSS Similarity Theorem:
If the sides of a two triangles are in proportion then, the triangles are similar.
Theorem: Triangle Proportionality Theorem:
If a line parallel to one of the sides of a triangle intersects the other two sides then, the line divides those sides proportionally.
If three parallel lines intersect two transversals then, the parallel lines divide the transversal proportionally.
Theorem: The Triangle Angle-Bisector Theorem:
If a ray bisects an angle of a triangle, then it divides the opposite side of the angle into segments proportional to the other two sides.
Ratios, Proportions and Similarities
The ratio of one number to another number is the quotient of the first number divided by the second number. The quotient is expressed in simplest forms.
Proportions are equations stating that the ratios are equal.
Polygons are similar if that their vertices can paired so that:
a) Their corresponding vertices are congruent.
b) Their corresponding sides are in proportion.
The length ratio of the legs of a right triangle is called the tangent ratio.
Consider the right triangle:
The tangent ratio is defined as:
The tangent of an angle depends only on the measure of the angle. There is a chart that shows the values of trigonometric functions for angles between 0Â° and 90Â°.
given the triangle:
Find the angle q.
Find the opposite side with q = 35Â°
From the right triangle:
The sine ratio is defined as:
consider the right triangle:
Find the length of x.
Angles and Segments
Angles and Segments
Inscribed angles are angles whose vertex is in the circle and the sides contain chords of the circle.
The measure of an inscribed angle is equal to half of the measure of its intercepted arc.
If two inscribed angles intercept the same arc, then the angles are congruent.
If a quadrilateral is inscribed in a circle, then the opposite angles of the quadrilateral are supplementary.
An angle inscribed in a semicircle is a right angle.
The measure of the angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.
The measure of an angle formed by two intersecting chords inside a circle is equal to half the sum of the measures of the intercepted arcs.
A secant is a line that intersects a circle in two points.
The measure of an angle made by two secants or two tangents or a secant and a tangent line drawn from a point outside the circle is equal to half the difference of the measures of the intercepted arcs.
Circles and Length of Segments:
When two chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
When two secant segments are drawn to a circle from a point outside the circle, the product of the lengths of one secant segment and its outer segment equals the product of the lengths of other secant segment and its outer segment.
When a secant segment and a tangent segment are drawn to a circle from a point outside the circle, the product of the length of the secant segment and its outer segment is equal to the square of the length of the tangent segment.
Tangent, Arcs, Chords and Circles
A circle is a set of points in a plane that are equidistant from a fixed point. A circle is denoted by .
The fixed point is called the center and the distance from the fixed point to the set of points is the radius.
A segment from the center to the point on the circle is also called the radius.
A segment that joins two points on a circle is called a chord.
A chord that passes through the center is called a diameter.
A secant is a line that contains a chord.
Diameter of a circle is twice the radius.
Congruent circles are circles that have congruent radii.
Concentric circles are circles that have the same center, and lie on the same plane.
A sphere is a set of points that are equidistant from a fixed point.
The terms used for circles are also used for spheres.
If a circle is drawn around a polygon, where the vertices of the polygon is touching the circle, it is said that the circle is circumscribed about the polygon.
If a polygon is drawn inside a circle where the vertices of the polygon is touching the circle then it is said that the polygon is inscribed in a circle.
A tangent of a circle is a line in the same plane as the circle and the line intersects the circle at exactly one point, called the point of tangency.
If a line is tangent to a circle then the radius is perpendicular to the line at the point of tangency.
Tangents to a circle from a given point are congruent.
If a line is perpendicular to the radius of a circle at the radius' outer endpoint then the line is tangent to the circle.
A line that is tangent to two coplanar circle is called a common tangent.
Circles are tangent to each other when both circles are tangent to the same line at the same point
An arc is an unbroken part of a circle.
Consider the circle:
If EF is the diameter then the arcs formed are semi circles.
The arc EF called a minor arc is formed by the interior ÃECF and the points on the between point E and F.
The remaining part of C and points E and F is called the major arc, denoted as EGF. Major arcs and semicircles are named with three points on the circle.
The central angle of a circle is an angle with its vertex at the center of the circle. The central angle of an arc is the central angle of a circle with the endpoints of the angle intersects a minor arc.
The central angle of is ÃECF.
The measure of a minor arc is the measure of its central angle.
The measure of a semicircle will always be 180Â°, and the measure of major arcs will always be larger than 180Â°.
Arcs having a single common point are adjacent non overlapping arcs.
Postulate: Arc Addition Postulate.
The measure of the arc formed by two non overlapping adjacent arcs is the sum of the measures of their central angles.
In congruent circles or in the same circle, two minor arcs are congruent if and only if their central angles are congruent.
Arcs and Chords:
In congruent circles or in the same circle:
1. Congruent arcs have congruent chords.
2. Congruent chords have congruent arcs.
A diameter that is perpendicular to a chord bisects that chord and the arc in the chord.
In congruent circles or in the same circles:
1. Chords that are equally distant from the center (or centers) are congruent.
2. Congruent chords are equally distant from the center (or centers).a
Concurrent lines are lines that intersect in one point.
The bisectors of angles of a triangle intersect at a point that is equidistant from the three sides of the triangle.
The perpendicular bisectors of the sides of a triangle intersect at a point that is equidistant from the three vertices of the triangle.
The lines that contain the altitudes of a triangle intersect at one point.
The medians of a triangle intersect at a point that is two thirds the distance from a vertex to the midpoint of the side opposite the vertex.
A locus is a figure that is a set of all points, and only those points,
that satisfy the one or more conditions that is stated.
The locus of a set of points in a plane that is 1cm. away from point C.
The figure is a circle with radius 1cm.
Area of Polygons:
Area of a polygon means the polygon itself and its interior.
Postulate: The Area of a Square.
The area of a square is the square of the length of a side.
( A = sÂ²)
Postulate: Area Congruence Postulate.
If two figures are congruent then their areas are equal.
Postulate: Area Addition Postulate.
The area of a region is the sum of the areas of its non-overlapping parts.
The area of a rectangle equals the product of the length of its base and the length of its height.
( A = bh)
Given a a circle, a regular polygon can inscribed in the circle, no matter how many sides the polygon have.
Also, given a regular polygon of any number of sides, a circle can be circumscribed about the regular polygon.
The center of a polygon is also the center of circumscribed circle.
The radius of a regular polygon is the distance from the center to a vertex of the polygon.
The central angle of a regular polygon is the angle formed by two radii drawn from two consecutive vertices.
The apothem of a regular polygon is the perpendicular distance from the center to a side of the polygon.
The area of a regular polygon is equal to the perimeter of the polygon and half the apothem. ( A = Â½ ap)
Area Of Triangles and Parallelograms
The area of a parallelogram is equal to the product of it base and
height.( A = bh)
The area of a triangle is the product of half the length of its base and
the length of the height.
The area of a rhombus equals half the product of its diagonals.
( A = ½ d1 d2 )
Theorem: The Area of Trapezoids:
The area of a trapezoid is equal to the product of half its height and
the sum of bases.
( A = ½ h( b1 + b2 ))
Area of a Circle:
p = 3.141592653589793237462643383279
The circumference of a circle is equal to the product of two times p and the radius.
( C = 2 pr )
The Area of a Circle is equal to the product of p and the square of the radius.
( A = pr²)
Area of Sectors and Arc Length:
Sector is a region that is bounded by two radii and an arc of the circle.
Area of a sector:
x = central angle:
Arc Length :
The bases of a prism are congruent and lie on parallel planes.
The altitude of a prism is a segment that joins the two bases and are perpendicular to the bases.
Lateral faces are not bases for prisms.
Lateral edges are the intersection segments of adjacent lateral faces.
The lateral faces of prisms are parallelograms, if the faces are rectangles, then the prism is called a right prism, otherwise the prism is called an oblique prism.
The lateral area (L.A.) of a prism is the sum of the areas of the lateral faces.
The total area (T.A.) of a prism is the sum of the areas of the lateral faces and the areas of the bases. ( T.A. = L.A. + 2B )
The lateral area of a right prism is the product of the perimeter of the
base and the height of the prism.( L.A. = ph)
The volume of a right prism is the product of the area of the base and the height of the prism.( V = Bh)
Point V in the figure is the vertex of the pyramid.
The segment perpendicular to the base from the vertex is the altitude and its length is the height h.
Regular Pyramids have:
The lateral area of a pyramid can be found in two ways:
Cylinder and Cones:
Cylinders are like prisms, the only difference is that instead of having polygons for a bases it has a circles for a bases. In a right cylinder, the perpendicular segment that joins the two circular bases at its centers is called the altitude. The length of the altitude is called the height of the cylinder, denoted as h. The radius of the base is also the radius of the cylinder.
Cones are similar to pyramids, The only difference is that the instead of having a polygon for a base the base of a cone is a circle. In a right cone, the altitude of a cone has a length called the height, and is a segment that is perpendicular to the base from the vertex to the center of the base. The slant height of a cone is the segment from the vertex to the edge of the base.
Volumes of Cylinders and Cones:
Volume of Cylinders:
V = Bh = prÂ²h
Volume for Cones:
Lateral Area of Cylinder:
L.A. = ph = 2prh
Lateral Area of Cones:
L.A. = prl
The Area and Volume of the Sphere:
A = 4pr
Areas and Volumes of Similar Solids:
If a : b is the scale factor of two figures then:
Theorem 11.2 The Equation of a Circle.
The equation of a circle with its center at (a,b) and a radius r is:
r²= (x - a)² + (y - b)²
Slope of a line:
If the slope of two lines are equal then the line are parallel.
If the product of the slopes of two non-vertical lines is -1 then the
lines are perpendicular.
Theorem: The Standard Equation of a Line.
Any equation that can be written as ax + by = c, where a and b are
nonzero numbers, is a line.
Theorem: Point-Slope Form
The equation of a line with point and slope m is:
y -y1 = m(x - x1)
Theorem: The Slope-Intercept Form
The equation of a line with slope m and y-intercept b is:
y = mx + b
Translation or Glide Transformations:
A translation or glide, is an isometry that glides all the points of a figure the same distance.
If a transformation T maps any point (x,y) to (x+a,y+b) then the transformation T is a translation.
Rotations are isometries.
Rotations are denoted by and the number of degrees of rotation.
The number of degrees is between 0 and 360, counter-clockwise rotation implies positive rotation and clockwise implies negative rotation.
Rotations are denoted by:
Â0,30 = P
Dilations are mappings that deal with similarities not congruence. Rotations, reflections, translations are all congruence mapping.
If | k | > 1 then the dilation is called an expansion.
If | k | < 1 then the dilation is called an contraction.
Products Of Mappings:
Products of mappings are denoted by AB: P Â® P" stated as the B followed by A of P to P".
The product of two isometries is an isometry.
The product of a reflection in two parallel lines is a translation. All the
points are translated through twice the distance between the lines.
The product of reflections in two intersecting lines is a rotation about
the point of intersection of the two lines. The angle of rotation is twice
the angle from the first line of reflection to the second line of
A product of reflections in perpendicular line is a half turn about the
point of intersection of the lines.
Inverse and Identity:
Identity transformation, denoted as I, is the mapping that maps every point to itself.
Identity and mapping is used in mapping just the same as it is used with numbers.
Inverse mapping is the transformation that is defined as ST = I.
Inverses are denoted as AÂ¯ Â¹. The same as the inverse of numbers.