Geometry EOC Review 2
Law of cosine
For △ABC, the Law of Cosines states that a2 = b2 + c2-2bccosA b2 = a2 + c2 - 2accosB c2 = a2 + b2 - 2abcosC
Bayes' Theorem
Given two events A and B with P (B) ≠ 0, P (AǀB) = P(BǀA) ⋅ P(A)/ P(B) . Another form is P (AǀB) = P(BǀA) ⋅ P(A)/ P(BǀA) ⋅ P(A) + P(BǀAc) ⋅ P(Ac).
You can extend the Multiplication Rule to three or more events. For instance, for three events A, B, and C, the rule becomes P (A ⋂ B ⋂ C) = P (A) ∙ P (B|A) ∙ P (C|A ⋂ B).
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You can represent dilations using the coordinate notation (x, y) → (kx, ky), where k is the scale factor and the center of dilation is the origin. If 0 < k < 1, the dilation is a reduction. If k > 1, the dilation is an enlargement.
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You know two tests for the independence of events A and B: 1. If P(A∣B) = P(A), then A and B are independent. 2. If P(A ∩ B) = P(A) ∙ P(B), then A and B are independent
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Intersection
A point where lines intersect.
Oblique prism
A prism that has at least one non-rectangular lateral face
Properties of similar figures
Corresponding angles of similar figures are congruent. Corresponding sides of similar figures are proportional.
Probability of Independent Events
Events A and B are independent if and only if P (A ∩ B) = P (A) · P (B)
Right cylinder
Has perpendicular to its center axis
Triangle Proportionality Theorem
If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally
Side-Angle-Side (SAS) Triangle Similarity Theorem
If two sides of one triangle are proportional to the corresponding sides of another triangle and their included angles are congruent, then the triangles are similar
Pythagorean Theorem
In a right triangle, the square of the sum of the lengths of the legs is equal to the square of the length of the hypotenuse. a^2+b^2=c^2
Lateral Area and Surface Area of Right Cylinders
The lateral area of a cylinder is the area of the curved surface that connects the two bases. The lateral area of a right cylinder with radius r and height h is L = 2πrh. The surface area of a right cylinder with lateral area L and base area B is S = L + 2B, or S = 2πrh + 2πr^2
Density
The amount of matter that an object has in a given unit of volume. The density of an object is calculated by dividing its mass by its volume.
Area Formula for a Triangle in Terms of its Side Lengths
The area of △ABC with sides a, b, and c can be found using the lengths of two of its sides and the sine of the included angle: Area =1/2bc sin A, Area =1/2ac sin B, or Area =1/2 ab sin C.
Conditional Probability
The conditional probability of A given B (that is, the probability that event A occurs given that event B occurs) is as follows: P (A|B) =P (A ⋂ B)/P (B)
Volume of a prism
The formula for the volume of a cube with edge length s is V = s^3
Volume of a prism
The formula for the volume of a right rectangular prism with length ℓ, width w, and height h is V = ℓwh
Lateral Area and Surface Area of a Right Cone
The lateral area of a right cone with radius r and slant height ℓ is L = πrℓ. The surface area of a right cone with lateral area L and base area B is S = L + B, or S = πrℓ + πr^2
Lateral Area and Surface Area of Right Prisms
The lateral area of a right prism with height h and base perimeter P is L = Ph. The surface area of a right prism with lateral area L and base area B is S = L + 2B, or S = Ph + 2B.
Geometric Means Theorem #2
The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
Geometric Means Theorem #1
The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.
Combinations
The number of combinations of n objects taken r at a time is given by n^C r =n!/r!(n - r)!
Cross section
The region of a plane that intersects a solid figure
Surface area
The total area of all the faces and curved surfaces of a three-dimensional figure.
Sine
opposite leg/ hypotenuse
Tangent angle
opposite leg/ adjacent leg
Lateral Area and Surface Area of a Regular Pyramid
The lateral area of a regular pyramid with perimeter P and slant height ℓ is L =1/2Pℓ. The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S =1/2Pℓ + B
Ambiguous case
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Similarity transformation
A transformation in which an image has the same shape as its pre-image
Circle similarity theorem
All circles are similar
Indirect measurements
Involves using the properties of similar triangles to measure such heights or distances
Geometric mean
Of 2 positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that x = √ab or x^ 2 = ab.
Lateral area
Of a prism is the sum of the areas of the lateral faces
The Addition rule
P (A or B) = P (A) + P (B) - P (A and B)
Volume of a pyramid
The volume V of a pyramid with base area B and height h is given by V= 1/3Bh
Volume of a cone
The volume of a cone with base radius r and base area B = πr^2 and height h is given by V =1/3Bh or by V =1/3πr^2h
Cosine
adjacent leg/ hypotenuse
Law of sines
sinA/a=sinB/b=sinC/c
A composite three-dimensional figure is formed from prisms and cylinders. You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure.
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Consider the proportion a/x =x/b where two of the numbers in the proportion are the same. The number x is the geometric mean of a and b
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Cross sections of three-dimensional figures sometimes turn out to be simple figures such as triangles, rectangles, or circles
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In a non-proportional dimension change, you do not use the same factor to change each dimension of a figure.
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In a proportional dimension change to a solid, you use the same factor to change each dimension of a figure
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In a proportional dimension change, you use the same factor to change each dimension of a figure.
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Pyramids that have equal base areas and equal heights have equal volumes
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Recall that the area of a triangle can be found using the sine of one of the angles. Area =1/2b · c · sin(A)
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Similarity transformations include reflections, translations, rotations, and dilations
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The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle
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The results of the Explore can be generalized to give a formula for permutations. To do so, it is helpful to use factorials. For a positive integer n, n factorial, written n!, is defined as follows. n! = n × (n - 1) × (n - 2) ×...× 3 × 2 × 1 That is, n! is the product of n and all the positive integers less than n. Note that 0! is defined to be 1. In the Explore, the number of permutations of the 7 objects taken 3 at a time is
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The right triangles you explored are sometimes called 45°-45°-90° and 30°-60°-90° triangles. In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg. In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg and the longer leg is √3 times as long as the shorter leg. You can use these relationships to find side lengths in these special types of right triangles
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The volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure
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To determine the independence of two events A and B, you can check to see whether P (A∣B) = P (A) since the occurrence of event A is unaffected by the occurrence of event B if and only if the events are independent.
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To find the missing side length of a right triangle, you can use the Pythagorean Theorem. To find a missing side length of a general triangle, you can use the Law of Cosines.
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To find the volume of a sphere, compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed.
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To show that two figures with all pairs of corresponding sides having equal ratio k and all pairs of corresponding angles congruent are similar, you can use similarity transformations.
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Two figures that can be mapped to each other by similarity transformations (dilations and rigid motions) are similar
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Two plane figures are similar if and only if one figure can be mapped to the other through one or more similarity transformations
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What if you are given the leg measures and want to find the measures of the acute angles? If you know the tan A, read as "tangent of ∠A," then you can use the tan-1 A, read as "inverse tangent of ∠A," to find m∠A. So, given an acute angle ∠A, if tanA = x, then tan -1 x = m∠A.
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When you know the length of a leg of a right triangle and the measure of one of the acute angles, you can use the tangent to find the length of the other leg.
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When you use the Law of Sines to solve a triangle for which you know side-side-angle (SSA) information, zero, one, or two triangles may be possible. For this reason, SSA is called the ambiguous case.
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Solving a right triangle means finding the lengths of all its sides and the measures of all its angles. To solve a right triangle you need to know two side lengths or one side length and an acute angle measure. Based on the given information, choose among trigonometric ratios, inverse trigonometric ratios, and the Pythagorean Theorem to help you solve the right triangle.
...You can use the distance formula as well as trigonometric tools to solve right triangles in the coordinate plane
Pythagorean Triples
A Pythagorean triple is a set of positive integers a, b, and c that satisfy the equation a 2 + b 2= c 2. This means that a, b, and c are the legs and hypotenuse of a right triangle. Right triangles that have non-integer sides will not form Pythagorean triples. Examples of Pythagorean triples include 3, 4, and 5; 5, 12, and 13; 7, 24, and 25; and 8, 15, and 17.
Set
A collection of distinct objects. Each object in a set is called an element of the set. A set is often denoted by writing the elements in braces
Right cone
A cone whose axis is perpendicular to the base
Oblique cylinder
A cylinder whose axis is not perpendicular to the bases
Net
A diagram of the surfaces of a three-dimensional figure that can be folded to form the three-dimensional figure
Trigonometric ratio (tangent, sine, cosine)
A ratio of two sides of a right triangle
Base of a regular pyramid
A regular polygon, and the lateral faces are congruent isosceles triangles
Regular pyramid
A regular polygon, and the lateral faces are congruent isosceles triangles
Directed line segment
A segment between two points A and B with a specified direction, from A to B or from B to A. To partition a directed line segment is to divide it into two segments with a given ratio.
Permutation
A selection of objects from a group in which order is important. For example, there are 6 permutations of the letters A, B, and C
Combination
A selection of objects from a group in which order is unimportant. For example, if 3 letters are chosen from the group of letters A, B, C, and D, there are 4 different combinations.
When you have a figure and its image after dilation, you can find the center of dilation by
Drawing lines that connect corresponding vertices. These lines will intersect at the center of dilation.
Right prism
Has lateral edges that are perpendicular to the bases, with faces that are all rectangles
Mutually Exclusive Events
If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B)
Converse of the Triangle Proportionality Theorem
If a line divides two sides of a triangle proportionally, then it is parallel to the third side
Side-Side-Side (SSS) Triangle Similarity Theorem
If the three sides of one triangle are proportional to the corresponding sides of another triangle, then the triangles are similar
Fundamental Counting Principle
If there are n items and a1 ways to choose the first item, a2 ways to select the second item after the first item has been chosen, and so on, there are a1 × a2 ×...×an ways to choose n items.
Angle-Angle (AA) Triangle Similarity Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar
Cavalieri's Principle
If two solids have the same height and the same cross-sectional area at every level, then the two solids have the same volume
Trigonometric Ratios of Complementary Angles
If ∠A and ∠B are the acute angles in a right triangle, then sinA = cosB and cosA = sinB. Therefore, if θ ("theta") is the measure of an acute angle, then sin θ = cos (90° - θ) and cos θ = sin (90° - θ)
Multiplication Rule
P (A ⋂ B) = P (A) ∙ P (B|A) where P (B|A) is the conditional probability of event B, given that event A has occurred
Probabilities of an Event and Its Complement
P(A) + P(A^c) = 1 The sum of the probability of an event and the probability of its complement is 1. P(A) = 1 - P(A^c)The probability of an event is 1 minus the probability of its complement. P(Ac) = 1 - P(A) The probability of the complement of an event is 1 minus the probability of the event.
Properties of dilations
Preserve angle measure, preserve betweenness, preserve collinearity, preserve orientation, map a line segment (the pre-image) to another line segment whose length is the product of the scale factor and the length of the pre-image, map a line not passing through the center of dilation to a parallel line and leave a line passing through the center unchanged
A probability experiment is an activity involving chance. Each repetition of the experiment is called a trial and each possible result of the experiment is termed an outcome. A set of outcomes is known as an event, and the set of all possible outcomes is called the sample space.
Probability measures how likely an event is to occur. An event that is impossible has a probability of 0, while an event that is certain has a probability of 1. All other events have a probability between 0 and 1. When all the outcomes of a probability experiment are equally likely, the theoretical probability of an event A in the sample space S is given by P(A) = number of outcomes in the event/number of outcomes in the sample space = n(A)/n(S) .
Law of Sines
The Law of Sines allows you to find the unknown measures for a given triangle, as long as you know either of the following: 1. Two angle measures and any side length --angle-angle-side (AAS) or angle-side-angle (ASA) information 2. Two side lengths and the measure of an angle that is not between them—side-side-angle (SSA) information
Permutations with Repetition
The number of different permutations of n objects where one object repeats a times, a second object repeats b times, and so on is n!/a!× b!×...
Permutations
The number of permutations of n objects taken r at a time is given by n^Pr =n!/(n - r)!
Conditional probability
The probability that event A occurs given that event B has already occurred is called the conditional probability of A given B and is written P(A|B)
Universal set
The set of all elements under consideration is the universal set, denoted by U
Empty set
The set with no elements is the empty set, denoted by ⌀ or { }
Surface Area of a Sphere
The surface area of a sphere with radius r is given by S = 4πr^2
Volume of a sphere
The volume of a sphere with radius r is given by V =4/3πr^ 3
Mutually exclusive events
Two events are mutually exclusive events if they cannot both occur in the same trial of an experiment. For example, if you flip a coin it cannot land heads up and tails up in the same trial. Therefore, the events are mutually exclusive.
Overlapping event
Two events are overlapping events (or inclusive events) if they have one or more outcomes in common.
Dependent events
Two events that fail either of these tests are dependent events because the occurrence of one event affects the occurrence of the other event.
Independent events
When the occurrence of one event has no effect on the occurrence of another event, the two events are called independent events.
Dilation
a transformation that can change the size of a polygon but leaves the shape unchanged. A dilation has a center of dilation and a scale factor which together determine the position and size of the image of a figure after the dilation
