New Rochelle High School Geometry Summer Assignment

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2 Evaluating Algebraic Expressions Evaluate Numerical Expressions - Numerical expressions often contain more than one operation. To evaluate them, use the rules for order of operations shown below. Order of Operations Step 1 Evaluate expressions inside grouping symbols. Step Evaluate all powers. Step Do all multiplication and/or division from left to right. Step 4 Do all addition and/or subtraction from left to right. Evaluate Algebraic Expressions - Algebraic expressions may contain more than one operation. Algebraic expressions can be evaluated if the values of the variables are known. First, replace the variables with their values. Then use the order of operations to calculate the value of the resulting numerical expression. Evaluate Numerical Expressions Evaluate Numerical Expressions a. [ + (1 ) ] Evaluate x + 5(y ) if x = and y = 1. [ + (1 ) ] = ( + 4 ) Divide 1 by. x + 5(y ) = ( + 16) Find 4 squared. + 5(1 ) Replace x with and y with 1. = (18) Add and (1 ) Evaluate. = 54 Multiply and (9) Subtract from Multiply 5 and 9. 5 Add 8 and 45. The solution is What is the value of 5xy if x and y? (1) 90 () 90 () 180 (4) 180. If y x 5x 6, what is the value of y when x? (1) () 4 () 4 (4) 4. Evaluate the expression [1 + (5 ) ]. 4. Evaluate the expression 5x (y + z) if x 6, y 8, and z. 5. Evaluate the expression (10x) + 100a if x and 4 a. 5

3 Solving Equations Multi-Step Equations - To solve equations with more than one operation, often called multi-step equations, undo operations by working backward. Reverse the usual order of operations as you work. Equations With Variables on Each Side - To solve an equation with the same variable on each side, first use the Addition or the Subtraction Property of Equality to write an equivalent equation that has the variable on just one side of the equation. Then solve the equation. Equations With Grouping Symbols - When solving equations that contain grouping symbols, first use the Distributive Property to eliminate grouping symbols. Then solve. Multi-Step Equations Equations With Variables on Each Side Solve 5x + =. Solve 5y 8 = y x + = Subtract from each side. 5y 8 y = y + 1 y Subtract y from each side. 5x = 0 Simplify. y 8 = 1 Simplify. 5x = Divide each side by 5. y = Add 8 from each side. x = 4 Simplify. y = 0 Simplify. Equations With Grouping Symbols y = 0 y = 10 Divide each side by. Simplify. Solve 4(a 1) = 10(a 5). 8a 4 = 10a + 50 Distributive Property 18a = 54 Simplify. 18a 8a a = 10a a Add 10a to each side. Divide each side by a 4 = 50 Simplify. a = Simplify. 18a = Add 4 to each side. 6. If 1x 8x 1 10, then x (1) () () 1 (4) 1 7. If x x 10, then x (1) 4 7 () 7 1 () 5 (4) 9 8. Solve x 1 9x Solve x 5 8x Solve 5 6 x x.

4 Factoring Factor x + bx + c - To factor a trinomial of the form x + bx + c, find two integers, m and p, whose sum is equal to b and whose product is equal to c. Factoring x + bx + c x + bx + c = (x + m)(x + p), where m + p = b and mp = c Factor x + 7x + 10 Factor x + 6x 16 In this trinomial, b = 7 and c = 10. Factors of 10, 5 7 Since + 5 = 7 and 5 = 10, let m = and p = 5. x + 7x + 10 = (x + 5)(x + ) Factor x 8x + 7 In this trinomial, b = 6 and c = 16. This means m + p is positive and mp is negative. Make a list of the factors of 16, where one factor of each pair is positive. In this trinomial, b = 8 and c = 7. Notice that m + p is negative and mp is positive, so m and p are both negative. Since 7 + ( 1) = 8 and ( 7)( 1) = 7, m = 7 and p = 1. Therefore, m = and p = 8. x 8x + 7 = (x 7)(x 1) x + 6x 16 = (x )(x + 8) 11. Which is a factor of Sum of Factors 1, y y 0? (1) y 6 () y 6 () y (4) y 1. Which product is a factored form of x x 1? x x x x Factors of 16 Sum of Factors 1, , 16 15, 8 6, 8 6 (1) () () x 6 x 1 (4) x 1 x 6 1. Factor x 8x Factor x 1x x 17x 60 4

5 Solving Quadratic Equations Solve Equations by Factoring - Factoring and the Zero Product Property can be used to solve many equations of the form x + bx + c = 0. Solve x + 6x = 7 x + 6x = 7 Original equation x + 6x 7 = 0 Rewrite equation so that one side equals 0. (x 1)(x + 7) = 0 Factor. x 1 = 0 or x + 7 = 0 Zero Product Property x = 1 x = 7 Solve each equation. The solution set is { 7, 1}, the roots are 7 and If x x 0, which is the greater of the two roots? (1) () () (4) 17. If x 11x 0 0, which is the smaller of the two roots? (1) 6 () 5 () 5 (4) Solve y y Solve x x Solve x x 10. 5

6 Slope Find Slope - The slope of a line is the ratio of change in the y-coordinates (rise) to the change in the x- coordinates (run) as you move in the positive direction. Find the slope of the line that passes through Find the value of r so that the line through (10, r) and (, 5) and (4, ). (, 4) has a slope of. 7 Let (, 5) = (x 1, y 1 ) and (4, ) = (x, y ). m = y y 1 x x 1 m = y y 1 x x 1 Slope formula 7 = 4 r 10 = 5 4 ( ) = 7 7 y =, y 1 = 5, x = 4, x 1 = 7 = 4 r 7 Slope formula m = 7, y = 4, y 1 = r, x =, x 1 = 10 Simplify. Simplify. ( 7) = 7(4 r) Cross multiply. = 1 14 = 8 7r Distributive Property 14 = 7r Subtract 8 from each side. = r Divide each side by What is the slope of line l shown in the. What is the slope of a line through points 4, 6,8? accompanying diagram? and (1) () 5 5 () (4) 5 (1) 5 () 5 () 5 5 (4) 5. Determine the slope of the line through the points A, and 1,7 B? 4. Determine the slope of the line through the points A 4, and 0,6 B? 5. If points, and, 5 x are on a line whose slope is 7, what is the value of x? 6

7 Equations of Lines Slope-Intercept Form y = mx + b, where m is the slope and b is the y-intercept Write an Equation Given the Slope and a Point Write an equation of the line that passes through (, 1) Write an Equation Given Two Points Write an equation of the line that passes through with a slope of 1. (1, ) and (, ). 4 The line has slope 1. Replace m with 1 and (x, y) with (, 1) 4 4 Find the slope m. To find the y-intercept, replace m with in the slope-intercept form. its computed value and (x, y) with (1, ) in the slope- y = mx + b Slope-intercept form intercept form. Then solve for b. 1 = 1 4 ( ) + b m = 1 4, y = 1, and x = m = y y 1 x x 1 1 = 1 + b Multiply. m = 1 Slope formula y =, y 1 =, x =, x 1 = 1 1 = b Add 1 to each side. m = Simplify. Therefore, the equation is y = 1 x 1. y = mx + b Slope-intercept form 4 = (1) + b Replace m with, y with, and x with 1. = + b Multiply. 4 = b Add to each side. Therefore, the equation is y = x What is an equation of the line in the 7. The line whose equation is x y 1 has accompanying diagram? (1) y x (1) slope = ; y-intercept = 6 () y x () slope = ; y-intercept = 6 () y x () slope = ; y-intercept = (4) y x (4) slope = ; y-intercept = 6 8. Write the equation of the line whose slope is and y-intercept is 5. 7

8 9. Write the equation of the line that passes through the point 0, and has a slope of 4? 0. Write the equation of the line that contains the points 1, and 1,1? 8

9 Graphing Lines Slope-Intercept Form y = mx + b, where m is the slope and b is the y-intercept Graph x 4y = 8. x 4y = 8 Original equation 4y = x + 8 Subtract x from each side. 4y = x y = 4 x Divide each side by 4. Simplify. The y-intercept of y = x is and the slope is. So graph the point (0, ). From this point, move up units 4 4 and right 4 units. Draw a line passing through both points. 1. Find the equation in slope-intercept form that describes the line graphed below. (1) y = x 1 () y = x + () y = x 1 (4) y = x +. Which is the graph of x 4y = 6? (1) () () (4). Write an equation in slope-intercept form for the graph shown. 4. Graph 1 y x on the answer page. 5. Graph y x 1 (show work here, place graph on the answer page). 9

10 Perimeter and Circumference Perimeter and Circumference The distance around a figure. Perimeter and circumference is measured in linear units. Find the perimeter or Circumference. Round to the nearest tenth. a. b. c. P = a + b + c P = l+ w C = πr = = () + () = π(5) = 1 in. = 10 ft. = 10π = 1.4 in. 6. A rectangle is 6. inches long and 1.7 inches wide. Find its perimeter. (1) 7.9 in. () in. () 15.8 in. (4) in. 7. If a circle has a radius of 6 inches, what is the circumference rounded to the nearest whole number? (1) 19 in. () 8 in. () 11 in. (4) 76 in. Find the perimeter of each figure

11 Area Area The number of square units needed to cover a surface. Area is measured in square units. Find the area. Round to the nearest tenth. a. b. c. A = 1 bh A = lw A = πr = 1 (4)() = ()() = π(5) = 6 in = 6 ft = 5π = 78.5 in 41. What is the area of a circle whose diameter is 14 centimeters? (1) cm () cm () 4.98 cm (4) cm 4. Find the area of the trapezoid. (1) 588 m () 94 m () 60 m (4) 15 m Find the area of each figure

12 Volume Volume The measure of space occupied by a solid. Volume is measured in cubic units. 46. Find the volume. 47. A storage container in the shape of a right (1) 576 cubic yards circular cylinder is shown in the accompanying () 416 cubic yards diagram. What is the number of cubic inches in () 100 cubic yards the volume of this container? (4) 16 cubic yards (1) () () 51. (4) Find the volume of each solid to the nearest tenth

13 Pythagorean Theorem The Pythagorean Theorem - The side opposite the right angle in a right triangle is called the hypotenuse. This side is always the longest side of a right triangle. The other two sides are called the legs of the triangle. To find the length of any side of a right triangle, given the lengths of the other two sides, you can use the Pythagorean Theorem. Pythagorean Theorem If a and b are the measures of the legs of a right triangle and c is the measure of the hypotenuse, then c = a + b. Find the missing length. c = a + b Pythagorean Theorem c = a = 5 and b = 1 c = 169 c = 169 c = 1 Simplify. Take the square root of each side. Simplify. The length of the hypotenuse is Given the right triangle below, what is the 5. Which set of three numbers represent the length of the hypotenuse? lengths of the sides of a right triangle? (1) cm () 1.5 cm (1) {4, 5, 6} () {6, 8, 10} () 4.4 cm () {8, 9, 10} (4) {9, 16, 5} (4) 19.0 cm 5. Solve for x. 54. Calculate the length of the hypotenuse, to the nearest tenth. 55. The accompanying diagram shows a kite that has been secured to a stake in the ground with a 0-foot string. The kite is located 1 feet from the ground, directly over point X. What is the distance, in feet, between the stake and point X? 1

14 Simplifying Square Roots Product Property of Square Roots - The Product Property of Square Roots and prime factorization can be used to simplify expressions involving irrational square roots. Product Property of Square Roots For any numbers a and b, where a 0 and b 0, ab = a b. Simplify = 6 5 Find the greatest perfect square root factor of 180 = 6 5 Product Property of Square Roots = 6 5 Simplify. 56. Simplify 1 (1) 6 () () (4) Simplify 00 (1) 0 10 () 10 () 10 0 (4) Simplify Simplify Simplify

15 NAME points out of Score - Geometry Summer Assignment Answer Page Directions - For Multiple Choice questions, place the choice number in the corresponding provided space. For Short Answer questions, leave the work in the previous pages and just place the final answer in the corresponding space. Evaluating Algebraic Expressions Page Solving Equations Page Factoring Page Solving Quadratic Equations Page Slope Page Equations of Lines Page Graphing Lines Page

16 Perimeter and Circumference Page Area Page Volume Page Pythagorean Theorem Page Simplifying Square Roots Page

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