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Ramapo- Indian Hills High School Summer Math Packet TUTORIAL Geometry CP/CPE To the students: The following tutorial is designed to prepare you for your Geometry CP/CPE course. You are advised to read all tutorials and answer the problems in the other packet over the summer. Please bring the completed problems on the first day of school in September. Thank you 0-4: Algebraic Expressions An expression is an algebraic expression if it contains sums and/or products of variables and numbers. To evaluate an algebraic expression, replace the variable or variables with known values, then use order of operations. Step 1: Evaluate expressions inside grouping symbols, such as ( ), { }, | |. or [ ] Step 2: Evaluate all powers. Step 3: Do all multiplication/division from left to right Step 4: Do all addition/subtraction from left to right. Example 1: Addition and Subtraction Algebraic Expressions Evaluate x – 5 + y if x = 15 and y = -7 x – 5 + y = 15 – 5 + (-7) Substitute = 10 + (-7) Subtract =3 Add Example 2: Multiplication and Division Algebraic Expressions Evaluate each expression if k = -2, n = -4, and p = 5 a. 2k + n 2(-2) + (-4) = 5-3 p-3 Substitute. b. -3(k2 + 2n) = -3[(-2)2 + 2(-4)] = -4 - 4 5-3 Multiply = -3[4 + (-8)] = -8 or -4 2 Subtract = -3(-4) or 12 Example 3: Absolute Value Expressions Evaluate 3|a – b| + 2|c – 5| if a = -2, b = -4, and c = 3 3|a – b| + 2|c – 5| = 3|-2 – (-4)| + 2|3 – 5| Substitute = 3|2| + 2|-2| Simplify = 3(2) + 2(2) Find Absolute Value = 10 Simplify 0.5: Linear Equations If each number is added or subtracted to each side from each side of an equation, the resulting equation is true. Example 1: Addition and Subtraction Solve each equation. a. x – 7 = 16 b. m + 12 = -5 x – 7 + 7 = 16 + 7 m + 12 – 12 = -5 – 12 x = 23 m = -17 If each side of an equation is multiplied or divided by the same number, the resulting equation is true. Example 2: Multiplication/Division Linear Equation Solve each equation. a. 4d = 36 4d 36 = 4 4 x=9 t b. - = -7 8 c. t -8(- ) = (-7)(-8) 8 5 3 5 ( )x = (-8) 3 5 3 3 x = -8 5 x=- t = 56 40 3 To solve equations with more than one operation, often called multi-step equations, undo the operations by working backwards. Example 3: Multi-Step Linear Equations Solve each equation. a. 8q – 15 = 49 b. 12y + 8 = 6y - 5 8q – 15 + 15 = 49 + 15 12y + 8 – 8 = 6y – 5 – 8 8q = 64 12y = 6y - 13 q=8 6y = -13 y=- 13 6 When solving equations that contain grouping symbols, first use the Distributive Property to remove the grouping symbols. Example 4: Multi-Step Linear Equations Solve 3(x – 5) = 13 3x – 15 = 13 3x = 28 x= 28 3 0.7 Ordered Pairs Points in the coordinate plane are known as ordered pairs. Ordered pairs are written in the form ( x, y) . The x-axis and y-axis divide the coordinate plane into four quadrants. The point of intersection of the axes is the origin. The origin is located at ( 0, 0) . Examples: 1. What point is located at (-3,-5) ? C 2. What are coordinates of B? (-3, 5) 3. What quadrant is point D in? IV 0.8: Systems of Linear Equations Two or more equations that have common variables are called a System of Equations. The solution of a system of equations in two variables is an ordered pair of numbers that satisfies both equations. A system of two linear equations can have zero, one, or an infinite number of solutions. There are three methods by which systems of equations can be solved: graphing, substitution, and elimination. Example 1: Graphing linear equations. Solve each system by graphing. Then determine whether each system has no, one, or infinitely many solutions. a. y = -2x + 8 y=x–1 The graphs appear to intersect at (3, 2). Check this estimate by replace 3 for x and 2 for y Check: y = -2x + 8 y=x-1 2 = -2(3) + 8 2=3-1 2=2✓ 2=2✓ The system has one solution (3, 2) b. y – 2x = 6 3y – 6x = 9 The graphs of the equations are parallel lines. Since they do not intersect, there are no solutions of this system of equations. Notice that the lines have the same slope but different y-intercepts. Equations with the same slope and the same y-intercepts have an infinite number of solutions. There are algebraic methods by which an exact solution can be found. One such method is Substitution. Example 2: Substitution y = -4x 2y + 3x = 8 Since y = -4x, substitute -4x for y Use y = -4x to find the value of y in the second equation. y = -4x æ 8ö y = -4 ç - ÷ è 5ø 32 = 5 2y + 3x = 8 2(-4x) + 3x = 8 æ 8 32 ö The solution is ç - , ÷ è 5 5ø -8x + 3x = 8 8 5 Sometimes adding or subtracting two equations together will eliminate one variable. Using this step to solve a system of equations is called Elimination. -5x = 8, so x = - Example 3: Elimination Use elimination to solve the system of equations 3x + 5y = 7 4x + 2y = 0 Either x or y can be eliminated. In this example, we will eliminate x 3x + 5y = 7 (Multiply by 4) 4x + 2y = 0 (Multiply by - 12x + 20y = 28 + (-12x) – 6y = 0 . 14y = 28 y=2 Now substitute 2 for y in either equation to find the value of x 4x + 2y = 0 4x + 2(2) = 0 4x + 4 = 0 4x = -4 x = -1 The solution is (-1, 2) 1.1: Points, Lines, and Planes Term Point Line Description A location It has neither shape nor size. Made up of points and has no thickness or width. Exactly 1 line through 2 points Names Point P Line m Line AB AB Ray Starts w/ a point called an endpoint and continues in opposite direction. Ray CD Part of a line w/ 2 endpoints. line segment ED Line Segment Plane CD ED A flat surface made up of points that extend without end; has no depth Plane G Plane ABC Collinear points: Points that are contained in the same line. Non-collinear points: Points that are not on the same line. Coplanar: Points and/or lines that are contained in the same plane. Picture The intersection of two lines is a point. The intersection of two planes is a line. 1.2 Linear Measure MEASURE LINE SEGMENTS A part of a line between two endpoints is called a line segment. Unlike a line, we can measure the length of a line segment because it has two endpoints. A segment with endpoints A and B can be named as AB or BA . The measure of AB is written as AB (without the segment bar above). Example 1: Find the length of MN using each ruler. Solution: The long marks are centimeters, and the shorter marks are millimeters. There are 10 millimeters for each centimeter. So, MN=34 mm. Example 2: Find the length of RS using each ruler. Solution: The long marks are inches and the short marks are quarter inches. So, RS = 1 ¾ inches. CALCULATE MEASURES In the figure, point N is between points A and B, but points R and P are not. NOTE: To be between does not mean midpoint. Notice in the diagram point N is closer to point A than point B. Point M is between points P and Q if and only if P, Q, and M are collinear (on the same line) and PM + MQ = PQ. Example 3: Find EG. Assume that the figure is not drawn to scale. Solution: EF is the measure of EG . Point F is between E and G. So, we can find EG by adding EF and FG. Example 4: Find AB. Assume that the figure is not drawn to scale. Solution: Point B is between A and C. So, Example 5: Find the value of a and XY if Y is between X and Z, XY = 3a, XZ = 5a – 4, and YZ = 14. Solution: Draw a figure to represent this information. Segments that have the same measure are called congruent segments. On AC , AB = BC = 3 cm. We can say that AB @ BC which we read as “segment AB is congruent to segment BC.” Slashes on the figure indicate which segments are congruent. Example 6: Determine whether each pair of segments is congruent. a) b) Solution: Yes, KJ @ HL Solution: We cannot since KJ = HL = 4 determine this from the given info Example 7: Determine whether XY and YZ are congruent. Solution: Since XY @ XZ , we know: XY = YZ 3x + 5 = 5x -1 3x + 5 - 3x = 5x -1- 3x 5 = 2x -1 5 +1 = 2x -1+1 6 = 2x 6 2x = 2 2 3= x x=3 Congruent segments. Substitution. Subtract each side by 3x. Simplify. Add 1 to each side. Simplify. Divide each side by 2. XY = 3x + 5 = 3(3) + 5 =14 9x 2 9(3) = = 13.5 2 YZ = Simplify. Reflexive Property. Since XY ¹ YZ , we can conclude that XY and YZ are not congruent. 1.3 Distance and Midpoints DISTANCE BETWEEN TWO POINTS Distance – between two points is the length of the segment with those points as its endpoints Distance Formula on a Number Line: PQ = |x2 – x1| or |x1 – x2| Example 1: Use the number line to find BE. Solution: Two find the distance between two points A and B in the coordinate plane, you can form a right triangle with AB as its hypotenuse and point C as its vertex. Then use the Pythagorean Theorem to find AB. Distance Formula (in Coordinate Plane): If P has coordinates (x1, y1) and Q has coordinates (x2, y2), then PQ = (x2 - x1 )2 + (y2 - y1 )2 Example 2: Find the distance between C(-4, -6) and D(5, -1). Solution: MIDPOINT OF A SEGMENT Midpoint – the point halfway between the endpoints of the segment If AB has endpoints at x1 and x2 on a number line, then the midpoint M of AB has coordinate: M= x1 + x 2 2 Example 3: The temperature on a thermometer dropped from a reading of 25o to –8o. Find the midpoint of these temperatures. Solution: x1 + x2 Midpoint Formula 2 25 + (-8) = x1 = 25, x2 =-8 2 13 = or 6.5 Simplify 2 M= You can find the midpoint of a segment on the coordinate plane by finding the average of the xcoordinates and of the y-coordinates of the endpoints. Midpoint Formula (in the Coordinate Plane): If PQ has endpoints at P (x1, y1) and Q(x2, y2) in the coordinate plane, then the midpoint M of PQ has coordinates: æ x + x 2 y1 + y 2 ö , ÷ M= ç 1 è 2 2 ø Example 4: Find the coordinates of M, the midpoint of ST , for S(-6, 3) and T(1, 0). Solution: Example 5: Find the coordinates of J if K(-1, 2) is the midpoint of JL and L has coordinates (3, -5). Example 6: Find the measure of PQ if Q is the midpoint of PR . Solution: Any segment, line, or plane that intersects a segment at its midpoint is called a segment bisector. In the figure at the right, M is the midpoint of PQ . Plane A, MJ , KM , and the point M are all bisectors of PQ . We say that they bisect PQ . 1.4 Angle Measure MEASURE AND CLASSIFY ANGLES A ray is a part of a line. It has one endpoint and extends indefinitely in one direction. Rays are named by stating the endpoint first and then any other point on the ray. We can name the ray to the right as MP or MO , but we cannot name it PM because we P is not the endpoint of the ray. Two collinear rays that share a common endpoint are called opposite rays. If two noncollinear rays have a common endpoint, they form an angle. The rays are the sides of the angle. The common endpoint is the vertex. When naming angles using three letters, the vertex must be the second letter. We can also name them using one letter, the vertex, when there is exactly one angle located at that vertex. We can also name angles by a single number if they are labeled. For example, the angle above can be named as ÐX, ÐXYZ, ÐZYX, or Ð3. An angle divides a plane into three distinct parts: 1.) Points Q, M, and N lie on the angle. 2.) Points S and R lie in the interior of the angle. 3.) Points P and O lie in the exterior of the angle. Example 1: a) Name all the angles that have B as a vertex. a) Ð5, Ð6, ÐABG, or ÐGBA b) Name the sides of Ð5. b) BG and BE or BF c) Write another name for Ð6. c) ÐEBD, ÐFBD, ÐDBF, or ÐDBE Angles are measured in degrees. We can use a protractor to measure angles. Below the degree measure of ÐDEF is 50, or we can say m ÐDEF = 50. The right angle is an angle whose measure is 90o. An acute angle is an angle whose measure is greater than 0o but less than 90o. An obtuse angle is an angle whose measure is greater than 90o but less than 180o. CONGRUENT ANGLES Angles that have the same measure are congruent angles. Since m ÐABC = m ÐFED, ÐABC @ ÐFED. The matching number of arcs on a figure also indicates congruent angles. A ray, line, or segment that divides an angle into two congruent angles is called an angle bisector. If YW is an angle bisector of ÐXYZ, then point W lies in the interior of ÐXYZ and ÐXYZ @ ÐWYZ . Just as with segments, when a line, segment, or ray divides an angle into smaller angles, the sum of the measures of the smaller angles equals the measure of the largest angle. So, in the figure to the right, mÐXYW + mÐWYZ = mXYZ . Example 3: In the figure, KJ and KM are opposite rays, and KN bisects ÐJKL . If mÐJKN = 8x -13 and mÐNKL = 6x +11, find mÐJKN. Solution: 1.5: Angle Relationships Adjacent Angles: two angles that lie in the same plane and have a common vertex and a common side (in other words, the angles share a side and are next to each other) For example: Ð1 & Ð2 are adjacent angles Linear Pair: adjacent angles that add up to 180 degrees; think the word LINE is in linear For example: Ð1 & Ð2 are adjacent angles that add up to 180 degrees Vertical Angles: 2 nonadjacent angles formed by 2 intersecting lines; they are also congruent For example: Ð1 & Ð2 and Ð3 & Ð4 are vertical angles Therefore: Ð[email protected] Ð2 and Ð3 @ Ð4 Complementary Angles: 2 angles that have a sum of 90 degrees For example: Ð1 is complementary to Ð2 and ÐA is complementary to ÐB Supplementary Angles: 2 angles that have a sum of 180 degrees For example: Ð3 is supplementary to Ð4 and ÐP is supplementary to ÐQ Perpendicular Lines( ^ ): lines, rays, or segments that form right angles 1.6: Two-Dimensional Figures Polygon: a closed figure made up of an finite number of segments called sides The vertex of each angle is the vertex of each polygon. You name a polygon by writing the vertices in consecutive order. Polygons can be convex: all diagonals can be drawn in the interior of the polygon OR concave: diagonals can be drawn both in the interior and exterior of the polygon equilateral polygon- polygon in which all sides are congruent equiangular polygon- polygon in which all angles are congruent regular polygon- both equilateral and equiangular Perimeter, Circumference, and Area