Geometry
Study:
Density formula
Undefined terms
Point - A location; has no dimension
Line - an infinite number of points expanding in opposite directions; has one dimension
Plane - A flat surface that extends infinitely; two dimensions
Defined terms
Line segment - a portion of a line with two endpoints
Ray - has one endpoint and extends infinitely in one direction
Vertex - the common endpoint of two rays
Angle - an angle is formed when two lines, segments, or rays intersect; angles are measured in degrees
Parallel lines - lines that lie in the same plane but never touch
Perpendicular lines - lines that intersect at right angle
Circle - a set of all points in a plane that are a given distance from a point, the center
Postulates - pieces of information that are accepted as facts; do not need to be proven.
Points postulate - through any two points there is exactly one line
Intersecting lines postulate - if two lines intersect, then they intersect in exactly one point. It is not possible for lines to intersect at more than one point
Intersecting planes postulate - if two distinct planes intersect, then they intersect in exactly one line. Their intersection creates a line.
Coplanar points postulate - through any three non collinear points, there is exactly one plane. If you have three points that are not on the same line, they will still lie on the same plane. Coplanar figures are two dimensional and have length and width.
Theorems - pieces of information that seem true but must be proven using postulates. Postulates, definitions, and undefined terms are used to prove theorems true.
Conditional statements - a statement that can be written in if then form. The hypothesis immediately follows the word if. The conclusion is the phrase following then.
Converse - formed by switching hypotheses and conclusion
Inverse - negate the hypothesis and conclusion
Contra positive - negate both statements and switch the hypothesis and conclusion. If the conditional statement was true, the contrapositive will be true.
Biconditional statements - when a conditional statement and its converse are true, they can be written as a single biconditional statement that qualifies as a definition. Written with “if and only if”
Proofs
Reflexive property of congruence - shows that something is congruent to itself
Transitive property of congruence - if a = b and b = c then a = c
Segment addition postulate - if B is between A and C, the measure of line segment AB plus line segment BC is equal to AC
Definition of a midpoint - a point that divides a segment into two equal parts and is halfway point in the given line segment
Angle addition postulate - When two smaller angles form to make a larger angle, the sum of the measures of the smaller angles will equal the measure of the larger angle.
Definition of congruence - two figures or objects are congruent if they have the same shape and size
Vertical angles theorem - vertical angles are congruent
Corresponding angles theorem - if a transversal intersects two parallel lines, then corresponding angles are congruent.
Alternate angles theorem - if a transversal intersects two parallel lines, then alternate interior angles are congruent
Alternate exterior angles theorem - if a transversal intersects two parallel lines, then alternate exterior angles are congruent
Same side interior angles theorem - if a transversal intersects two parallel lines, then same side interior angles are supplementary
Angle relationships
Complementary angles - angles that add up to 90. Adjacent or non adjacent
Supplementary angles - angles that add up to 180. Adjacent or non adjacent
Adjacent angles - two angles that share a side but have no interior points in common
Linear pair - two adjacent, supplementary angles
Vertical angles - opposite angles that are created when two lines intersect. These angles share a vertex but do not have sides or interior points in common.
Transformations
A transformation in mathematics is when a figure changes in some way. Its size, orientation, and/or location may change. Transformations can be rigid, nonrigid, or a combination of both.
A rigid motion is the action of taking an object and moving it to a different location without altering its shape or size.
Vectors - transformations can be represented using vectors. Vectors describe objects in motion. Vectors have a fixed length. They begin at the initial point and the tip of the arrow is the terminal point.
A reflection is a transformation where the mirror image of a figure is shown directly opposite, or across a line of reflection. The image and premise are congruent. Reflections are rigid motions.
X axis reflection: reflecting across the line y = 0. Rule: (x,-y)
Y axis reflection: reflecting across the line x = 0. Rule: (-x, y)
Line y = x reflection: rule: (y, x)
Reflecting across vertical lines: equation will always be x =
Reflecting across horizontal lines: equation will always be y =
Rotation: a transformation in which a figure turns around a fixed point. The fixed point is the center of rotation.
If you create an angle with the vertex at the origin, one ray passing through a point on the pre-image, and the other ray passing through its corresponding point on the image, the angle of rotation is shown with the appropriate degree measure
Rotation rules:
90 clockwise/270 counter clockwise: (y, -x)
270 clockwise/ 90 counter clockwise: (-y, x)
180: (-x, -y)
Congruency
Congruent figures have corresponding congruent side lengths and congruent angle measures. Orientation has nothing to do with congruence.
Triangle congruence postulates
SSS: if the corresponding sides of one triangle are congruent to the corresponding sides of another triangle, then the triangles are congruent.
SAS: if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent
ASA: if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the the triangles are congruent
AAS: if two angles and a non-included side of one triangle are congruent to the corresponding two angles and corresponding non included side of a second triangle, then the two triangles are congruent.
HL: if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and the corresponding leg of another right triangle, the the two right triangles are congruent.
NOT CONGRUENCE POSTULATES: aaa (at least one side is required to prove congruence. ssa is also not a congruence postulate.
Reflexive property shows that a shared side is a congruent side
Triangle proofs
Equilateral: three congruent sides and angles
Scalene: no congruent sides; no congruent angles
Isosceles: at least two congruent sides; two congruent angles. Symmetrical
Triangle sum theorem: the measures of the angles add up to 180.
Triangle inequality theorem: the sum of the sides of a triangle is always larger than the length of the third side
Hinge theorem: if two triangles have two congruent corresponding sides, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third, opposite side of the first triangle will be larger than the third, opposite side of the second triangle
Isosceles triangle theorem: if two sides of a triangle are congruent, the the angles opposite those sides are congruent.
Midsegment of a triangle theorem: a segment connecting the midpoints of two sides of a triangle is parallel to the third side and its length is equal to half the length of the third side.
The centroid theorem: the centroid of a triangle is located two thirds of the way from a vertex to the midpoint of the opposite side of the triangle.
Special segments and centers of a triangle
Centroid: medians are drawn from the midpoint of each side to the opposite vertex. The centroid represents the point of intersection between the three medians. The centroid will always lie inside the triangle.
Orthocenter: altitudes are drawn from each vertex of the triangle and intersect the opposite side, which may be extended, at a 90 angle. The ortho enter represents the point of intersection between the three altitudes. It can appear inside, outside, or on the triangle. It will fall outside for obtuse triangles and on the actual triangle for right triangles.
Incenter: angle bisectors are drawn from each vertex, splitting the angle at the vertex in two, to the opposite side. The incenter represents the point of intersection between the three angle bisectors. It will always lie inside the triangle.
Circumcenter: perpendicular bisectors intersect each side of the triangle at a 90 angle at that side’s midpoint. The circumvented represents the point of intersection between the three perpendicular bisectors. It can appear inside, outside, or on the triangle. It will fall outside for obtuse triangles and on the actual triangle for right triangles.
Quadrilateral proofs
A polygon is a closed figure with three or more straight sides.
Quadrilaterals are polygons that have four sides and four vertices
Parallelograms: both pairs of opposite sides are congruent and parallel. The diagonals bisect each other. Both pairs of opposite angles are congruent. Consecutive angles are supplementary.
Trapezoid midsegment theorem: the midsegment of a trapezoid is parallel to both bases. The length of the midsegment is one half the sum of the measures of the bases of the trapezoid
Dilations
In a dilation, side lengths change but stay proportional, angle measures stay the same.
Center of dilation- a fixed point around which a figure is dilated. To find it, plot a line between the corresponding points of the pre-image and the image
The area of the dilated figure is the area of the pre-image, multiplied by the square of the scale factor
The volume of the dilated figure is the volume of the pre image multiplied by the cube of the scale factor
Similarity
Similar polygons are polygons that have congruent angles and corresponding sides that are proportional to one another
Scale factor is image/pre image
Angle-Angle similarity postulate: if two corresponding angles of two or more triangles are congruent, the triangles are similar
Order is important when talking about congruency/similarity
Side angle side similarity postulate: if two or more triangles have one pair of corresponding, congruent angles and the sides that create these angles are proportional, then the triangles are similar
Side side side similarly postulate: if two or more triangles have three corresponding proportional sides, then the triangles are similar
Triangle proportionality theorem: if a segment is parallel to one side of a triangle and intersects the other two sides of the triangle the segment divides the sides of the triangle proportionally.
Pieces of a right triangle similarity theorem: if an altitude is drawn from the right angle of a right triangle, the two smaller triangles created are similar to one another and to the larger triangle.
Converse of the Pythagorean theorem: if the sum of the squares of the shorter sides is equal to the square of the longest side, then the triangle has a right angle
Pythagorean theorem: if a triangle has a right angle, then the sum of the square of the shorter sides is equal to the square of the longest side.
Coordinate geometry
Parallel lines have the same slope
The equations of perpendicular lines are special because the slopes of the lines are opposite reciprocals
To determine the length of a line, use the distance formula
Midpoint formula is used to find the midpoint between two points on a line segment
To find out if a triangle has a right angle, use the slope formula and check for perpendicular slopes. Use distance formula for side lengths
Parallelogram: distance formula- adjacent sides are not equal. Slope formula- adjacent sides are not perpendicular
Rhombus: distance formula- adjacent sides are equal. Slope formula- adjacent sides are not perpendicular
Square: distance- adjacent sides are equal. Slope- adjacent sides are perpendicular
Rectangle: distance- adjacent sides are not equal lengths. Slope- adjacent should be perpendicular
Slope: horizontal lines have a slope of zero, vertical lines have an undefined slope.
Splitting the distance: 1. Find the x-coordinate distance, multiply by the fraction to find the fraction of that distance. 2. Find the y-coordinate distance, multiply by the fraction to find the fraction of the distance. 3. Move. Add the original x-coordinate to the result of step one. Add the original y-coordinate to the result of step two.
Finding the distance that is at a ratio of 1:4 between two points is the same as splitting or partitioning the distance into 5 equal pieces and finding one of those pieces. (1/5)
Weighted averages: 1. Find the weight of each data point. 2. Multiply the weight by the associated value. 3. Add the results to calculate the weighted average.
Special Right triangles
To solve 45-45-90 triangles: if you know either leg, multiply by the square root of two to get the hypotenuse. If you know the hypotenuse, divide by the square root of two to get either leg.
To solve 30-60-90 triangles: if you know the hypotenuse, divide by two to get the short leg. If you know the short leg, multiply by two to get the hypotenuse. If you know the short leg, multiply by the square root of three to get the long leg. If you know the long leg, divide by the square root of three to get the short leg.
Solving right triangles
Every right triangle has three parts: two legs and a hypotenuse. The hypotenuse is always across from the right angle. The opposite leg is across from the angle being focused on, the adjacent leg is the one next to the angle being focused on.
For similar triangles, the trigonometric ratios will always be the same.
The tangent of a given angle is equal to the slope of the hypotenuse
Unit circle: circle whose radius is one and whose center is at the origin (0,0). The equation is x²+y²=1. Theta represents the angle Ø. Starting from (1,0), move counterclockwise until the angle that is formed between your position, the origin, and the positive x-axis is equal to Ø.
Sin Ø = y/1 = y. Cos Ø = x/1 = x. Tan Ø = y/x
Angle:
30° : cos ø: √3/2. Sin ø: ½
45° : cos ø: √2/2. Sin ø: √2/2
60° : cos ø: 1/2. Sin ø: √3/2
For complementary angles, the sine of one angle is equal to the cosine of the other.
Inverse ratios are used to find angles instead of sides.
Reciprocal functions: reciprocal of sine is csc. 1/sinø. The reciprocal of cos is sec. 1/cosø. The reciprocal of tan is cot. 1/tanø.
Angle of elevation: an angle of elevation is the angle at which an observer must direct their line of sight in an upward motion to view an object
Angle of depression: an angle of depression is the angle at which an observer must direct their line of sight in a downward motion to view an object.
The angle of elevation and the angle of depression are congruent bc they are alternate interior angles between parallel, horizontal, lines of sight.
Unit six
Radius: a straight line drawn from the center of the circle to the edge of the circle
Diameter: a straight line drawn from edge to edge of a circle passing through the center
Chord: any straight line drawn between two points that lie on a circle
The volume of a cylinder is basexheight. The base is a circle, so the area of it is found by πr2
The capital B in formulas means area of the base
Cavalieri’s principle: if the area of the cross-section of two 3-D figures are congruent and the heights of the figures are also congruent, then it can be concluded that the volume of the two figures are congruent.
Cross-section: the shape we get when cutting straight through an object.
A cylinder is a 3-dimensional figure having two parallel bases that are congruent circles
In a right cylinder, the height or altitude can be drawn so that it connects to the centers of the circular bases.
In an oblique cylinder, the height or altitude cannot be drawn so that it connects the centers of the circular bases. Height is always represented by a perpendicular segment
A cone is a 3-dimensional figure with one vertex and a circular base.
In a right cone, the height or altitude can be drawn so that it connects to the vertex of the cone with the center of the base
In an oblique cone, the height or altitude cannot be drawn so that it connects to the vertex of the cone with the center of the base
A sphere is the set of all points in space that are a fixed distance from a given point called the center
If you have a pyramid and prism with the same height and equal bases, the pyramid can hold 1/3 the volume of the prism
When you know the scale factor between two similar solids, the ratio of the volumes can be found by cubing that scale factor. The ratio between the corresponding sides of two similar solids can be represented in general terms by a:b. The ratio of the volumes of similar solids can be represented by the ratio a³ : b³
Density is a ratio of mass to volume. Density formula: mass/volume
If number of people objects in something, that is represented with mass
A cross section of a 3 dimensional figure is the intersection of a solid by a plane. Cross sections can be taken parallel to the base of a 3 dimensional figure, perpendicular to the base, or at a diagonal to the base.
When a plane passes through a 3 dimensional figure to create a cross section that is parallel to the base, the resulting 2 dimensional shape of the cross section is the same as the shape of the base
A 2 dimensional figure can be rotated, or spun, about an axis quickly to create a 3 dimensional figure.
Rectangles create cylinders
Triangles create cones
Semi circles create spheres
Nothing can create a pyramid because the base is not round
If the object is not touching the line it is rotated around, there will be a hole in the middle
A net drawing is a two dimensional pattern of a three dimensional figure that can be folded to form the actual three dimensional figure
Rectangle and two circles for a cylinder
Small circle and a larger quarter circle for a cone
Elongated ellipses for a sphere
4 equal sized triangles surrounding a square for a pyramid
2 congruent triangles and 3 congruent rectangles for a triangular prism
Surface area formulas
Square pyramid: b² + 2bs (b is base length, s is slant height)
Triangle based pyramid and rectangle based pyramid: A + ½ ps (A is base area, p is perimeter of base, s is slant height)
Cone: πr2 + πrs (s is slant height)
Cylinder: 2πr2 + 2πrh
Sphere: 4πr2
Rectangular prism: 2lw + 2wh + 2lh
Triangular prism: bh + pl (b is length of base, h is height of base, l is length of prism (between bases, p is perimeter of base)
Slant height is measured in the outside or lateral surfaces of the 3-dimensional figures.
Circles
A circle is the set of all points in a plane that are the same distance away from a fixed point
The radius is the distance between the center of a circle and a point on the circle
A chord is a segment on the interior of a circle with endpoints on the circle
Diameter is a type of chord that passes through the center of the circle
Circumference is the distance around the circle from one point back to the same point
Minor arc is an arc measuring less than 180°. Named with two letters
Major arc is an arc measuring more than 180°. Named with 3 letters
A secant is a line that passes through a circle intersecting it at two distinct points.
Tangent is a line that intersects the circle in exactly one point, it is perpendicular to the radius.
Concentric circles are two distinct circles that share a common center
A central angle is an angle on the interior, or inside, of a circle with its vertex at the circle’s center.
Adjacent arcs are arcs that share a common point
The arc addition postulate: the measure of an arc created by two adjacent arcs may be found by adding the measures of the two adjacent arcs.
Non adjacent arcs are arcs that do not have a common point on the circumference of the circle
Congruent arcs: two arcs are congruent if the central angles that intercept them are also congruent
The equation of angles with vertices inside the circle: ½ (arc1 + arc2)
Inscribed angles: when an angle is in the interior of a circle and its vertex is a point on the circle, the angle is called an inscribed angle. The measure of an inscribed angle is half the measure of its intercepted arc
An exterior angle to a circle may be found by subtracting the measure of the two arcs intercepted by the angle and dividing by two. exterior angle: ½ (arc1 - arc2)
Interior angle to a circle theorem: the measure of an angles created by chords, secants, or a combination on the interior of a circle is equal to half the sum of the arcs it and its vertical angles intercept
Inscribed angle theorem: the measure of an inscribed angle is equal to half the measure of its intercepted arc
Central angle theorem: the measure of a central angle is equal to the measure of the arc it intercepts
Exterior angle to a circle theorem: if two secants, two tangents, or a secant and a tangent intersect outside a circle, the measure of the created angle between them is one half the absolute value of the difference of the measures of their intercepted arcs
If two chords are congruent, then the central angles that create them are congruent
If two arcs are congruent then the chords that create them are congruent.
If two chords are the same distance from the center then they are congruent
If the diameter or the radius is perpendicular to a chord then it bisects the chord and its arc
Two chords that intersect is a circle have segments whose products are equal
If two tangents have a common endpoint outside of a circle then the distances from the point of tangencies to the point where the tangents intersect each other are equal
If two secants intersect outside of a circle then the product of the measures of one secant segment and its exterior secant segment is equal to the product of the measures of the other secant segment
If a tangent and a secant intersect outside of the circle then the square of the measure of the tangent segment is equal to the product of the measure of the secant segment and its exterior secant segment
A circumscribed circle is a circle that surrounds a polygon, while an inscribed circle is a circle contained within a polygon where each side of the polygon represents a tangent to the circle. In order to construct an inscribed circle to a given polygon, you must first know how to find its incenter
Angle bisectors lead to finding the incenter. The height of a triangle represents the radius of the circle. After finding the incenter, one must construct a perpendicular line to at least one side of the triangle in order to construct the inscribed circle
Steps for constructing an inscribed circle within a triangle: 1. Start with a triangle. 2. Construct two angle bisectors. 3. Find the radius. Once the incenter is found, construct a perpendicular line through the incenter to one of the sides. Extend the side chosen. 4. With the compass on the incenter, create two arcs on the line. Label the pints of intersection. Place the tip of your compass on one of the points and create an arc somewhere to the left of the other point. 5. Without changing the width of the compass, move it to the lower point and construct an intersecting arc to the previous arc. 6. Use the straightedge to draw a ray through the points. Mark and label the point of intersection between the ray and a segment. 7. To construct a circle with this radius, place the tip of your compass on the bottom point. Adjust the width so it is equal to the length of a segment.
Circumscribed vs inscribed circles
Circumscribed circles go around the triangle. Constructed with perpendicular bisectors. Circumcenter is equidistant from the three vertices of the triangle
Inscribed circles go inside the triangle. Constructed with angle bisectors. The I center is equidistant from the sides of a triangle.
Inscribed angles
When an inscribed angle exists in a circle its measure is half the measure of the arc and its intercepts.
Congruent inscribed angles theorem: two or more distinct inscribed angles that intercept the same arc, or congruent arcs, are congruent
Inscribed quadrilaterals theorem: Opposite angles of an inscribed quadrilateral are supplementary
Equations of circles: all circles have an equation. From its equation, we can determine the center and the radius of the circle. The equation of the circle is derived from the Pythagorean theorem. Equation of a circle where (h,k) is the center and r is the radius. (X-h)² + (y-k)² =r²
All circles are similar.
The arc length of a circle is the distance between two points on a circle.The value of the arc length of a circle can be determined by taking the formula for the circumference of a circle and multiplying it by the central angle divided by 360°
A radian is the measure of an intercepted arc of a central angle whose length is equal to the radius of the circle
Converting degree measures to radians: 2π(x)/360
the are of a sector of a circle is the 2 dimensional space inside the circle shaped area bound by a central angle. The area of the sector is proportionate to the lengths of the radii
πr² (x/360)