JANE LONG ACADEMY HIGH SCHOOL MATH SUMMER PREVIEW PACKET SCHOOL YEAR. Geometry
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1 JANE LONG ACADEMY HIGH SCHOOL MATH SUMMER PREVIEW PACKET SCHOOL YEAR Geometry STUDENT NAME: THE PARTS BELOW WILL BE COMPLETED ON THE FIRST DAY OF SCHOOL: DUE DATE: MATH TEACHER: PERIOD:
2 Algebra 1 Skills: Question # 1 Rectangle PQRS is graphed on the coordinate plane below What is the equation of the line that is perpendicular to PR through point S? A. y = 1 2 x B. y = 1 2 x 9 2 C. y = 2x 18 D. 2y = 4x 18 Question # 2 Line k is represented by the equation 2x + 3y = 5. Which equation below models a line that is parallel to line k? A. 9y 6x = 8 B. 2x + 3y = 5 C. 9x + 6y = 10 D. 2y 3x = 5
3 Examine the pattern below Question # 3 If the pattern continues, which expression below represents the total number of dots in the n th figure? A. n+4 B. 4n-1 C. 4n-3 D. 4n - 1 Question #4
4 Question # 5 Three of the vertices of parallelogram ABCD are given as A(2,1), B(8,5), and C(11,0). Which answer choice below could not be the slope of one of the sides or the slope of the diagonal containing the fourth vertex? A. Not enough information 5 B. 3 C. 2 3 D. 1 9 Question # 6 Determine the perimeter of the triangle below. ( ABC below) P =? Question # 7 Determine the slope of AB if Point A( -12, -5) and B(-4, -5) Question # 8 a) Determine the exact value of x, if A( x, -7) and B( -3, 12) are collinear points, and segment AB is perpendicular to the line passing through points C( -13, -2) and D(-7, -5) b) Explain the geometric term Collinear feel free to use the internet. Question # 9 a)determine the exact value of x, if A( x, -7) and B( -3, 12) are collinear points, and segment AB is parallel to the line passing through points C( -13, -2) and D(-7, -5)
5 b) Explain the difference between the following geometric terms Concave polygon and Convex polygon and draw an example of each. You must use the internet Questions #10 and #11: (Solve Problem 15b first)
6 Question #12 Use the data below to create a table and use the table to the solution as you learned in algebra 1. Question #13
7 Question #14 Question # 15a Question # 15b True or False? 1) All diagonals of any polygon connects two non-consecutive vertices. 2) All diagonals of any polygon are always inside the polygon.
8 Question #16 Question # 17 Question # 18 (Skills: Algebra 1 and Middle school geometry, Transformation) Use separate grid papers to draw each figure for the problem below ( each drawing must be on its grid paper) Sketch the region enclosed by the line 5x + 3y = 15, x = 0, and y = 0. a) Find the area of the region. b) Sketch and find the area of the figure formed when the region is reflected about the x-axis. c) Sketch and find the volume of the solid formed when the region is revolved around the x-axis. (hint: V = Bh = πr 2 h), ( Leave the answer in terms of π)
9 Question #19 a) Determine the equation of the line passing through points A and B. (Skills: Algebra 1) b) Determine the length of segment AC, using Pythagorean theorem (hint: make a right triangle on this grid were AC is the hypotenuse), (Skills: Middle school math) Question #20
10 The sections below are math reading sections: 1)Deductive reasoning Deductive reasoning is the process of showing that certain statements follow logically from agreed-upon assumptions and proven facts. 2)Inductive reasoning Inductive reasoning is the process of observing data, recognizing patterns, and making generalizations about those patterns. (See Problems 10 to 16 from your summer packet for 2015) Throughout this course you will use inductive reasoning. You will perform investigations, observe similarities and patterns, and make many discoveries that you can use to solve problems. Inductive reasoning guides scientists, investors, and business managers. All of these professionals use past experience to assess what is likely to happen in the future. When you use inductive reasoning to make a generalization, the generalization is called a conjecture.
11 Sometimes a conjecture is difficult to find because the data collected are unorganized or the observer is mistaking coincidence with cause and effect. Good use of inductive reasoning depends on the quantity and quality of data. Sometimes not enough information or data have been collected to make a proper conjecture. For example, if you are asked to find the next term in the pattern 3, 5, 7, you might conjecture that the next term is 9 the next odd number. Someone else might notice that the pattern is the consecutive odd primes and say that the next term is 11. If the pattern was 3, 5, 7, 11, 13, what would you be more likely to conjecture? Good use of deductive reasoning depends on the quality of the argument. Just like the saying, A chain is only as strong as its weakest link, a deductive argument is only as good (or as true) as the statements used in the argument. A conclusion in a deductive argument is true only if all the statements in the argument are true. Also, the statements in your argument must clearly follow from each other. Chapter 2; Discovering Geometry PDF Check Your Understanding by answering the questions below:
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13 3) Introduction to Logic Statements When we define and explain things in geometry, we use declarative sentences. For example, "Perpendicular lines intersects at a 90 degree angle" is a declarative sentence. It is also a sentence that can be classified in one, and only one, of two ways: true or false. Most geometric sentences have this special quality, and are known as statements. In the following lessons we'll take a look at logic statements. Logic is the general study of systems of conditional statements; in the following lessons we'll just study the most basic forms of logic pertaining to geometry. Conditional statements are combinations of two statements in an if-then structure. For example, "If lines intersect at a 90 degree angle, then they are perpendicular" is a conditional statement. The parts of a conditional statement can be interchanged to make systematic changes to the meaning of the original conditional statement. Based on the truth value (there are only two truth values, either true or false) of a conditional statement, we can deduce the truth value of its converse, contrapositive, and inverse. These three types of conditional statements are all related to the original conditional statement in a different way. GEOMETRY: LOGIC STATEMENTS Variations on Conditional Statements The three most common ways to change a conditional statement are by taking its inverse, its converse, or it contrapositive. In each case, either the hypothesis and the conclusion switch places, or a statement is replaced by its negation. The Inverse The inverse of a conditional statement is arrived at by replacing the hypothesis and the conclusion with their negations. If a statement reads, "The vertex of an inscribed angle is on a circle", then the inverse of this statement is "The vertex of an angle that is not an inscribed angle is not on a circle." Both the hypothesis and the conclusion were negated. If the original statement reads "if j, then k ", the inverse reads, "if not j, then not k." The truth value of the inverse of a statement is undetermined. That is, some statements may have the same truth value as their inverse, and some may not. For example, "A four-sided polygon is a quadrilateral" and its inverse, "A polygon with greater or less than four sides is not a quadrilateral," are both true (the truth value of each is T). In the example in the paragraph above about inscribed angles, however, the original statement and
14 its inverse do not have the same truth value. The original statement is true, but the inverse is false: it is possible for an angle to have its vertex on a circle and still not be an inscribed angle. The Converse The converse of a statement is formed by switching the hypothesis and the conclusion. The converse of "If two lines don't intersect, then they are parallel" is "If two lines are parallel, then they don't intersect." The converse of "if p, then q " is "if q, then p." The truth value of the converse of a statement is not always the same as the original statement. For example, the converse of "All tigers are mammals" is "All mammals are tigers." This is certainly not true. The converse of a definition, however, must always be true. If this is not the case, then the definition is not valid. For example, we know the definition of an equilateral triangle well: "if all three sides of a triangle are equal, then the triangle is equilateral." The converse of this definition is true also: "If a triangle is equilateral, then all three of its sides are equal." What if we performed this test on a faulty definition? If we incorrectly stated the definition of a tangent line as: "A tangent line is a line that intersects a circle", the statement would be true. But it's converse, "A line that intersects a circle is a tangent line" is false; the converse could describe a secant line as well as a tangent line. The converse is therefore a very helpful tool in determining the validity of a definition. The Contrapositive The contrapositive of a statement is formed when the hypothesis and the conclusion are interchanged, and both are replaced by their negation. In other words, the contrapositive of a statement is the same as the inverse of that statement's converse, or the converse of its inverse. Take the statement, "Long books are fun to read." Its contrapositive is "Books that aren't fun to read aren't long." The statement "if p, then q " becomes "if not q, then not p." The contrapositive of a statement always has the same truth value as the original statement. Therefore, the contrapositive of a definition is always true. For example, the statement "A triangle is a three-sided polygon" is true. Its contrapositive, "A polygon with greater or less than three sides is not a triangle" is also true.
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