Squares and Square Roots Algebra 11.1
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1 Squares and Square Roots Algebra 11.1 To square a number, multiply the number by itself. Practice: Solve (9) Squares and Square Roots are Inverse Operations. If =y then is a square root of y. Every positive number has two square roots. You will only need to indicate the positive square root of a number unless this symbol appears before the radical: Practice: Solve without a calculator , 000 The following square roots should be easy to calculate in your head. Double check by squaring your answer. Practice: Solve without a calculator. 1. 1, , Be careful and check your work on these , 000
2 Squares and Square Roots Algebra 11.1 Simplifying radical epressions. There are several easy rules you must know for simplifying square roots. Fractions: a a 9 b b Products: ab a b Practice: Solve without a calculator , (tricky... think!) Simplifying irrational radical epressions. Irrational numbers are non-terminating, non-repeating decimals. Some square roots cannot be simplified into integers, fractions, or decimals. Eample: Simplify Practice: Simplify
3 Squares and Square Roots Algebra 11.1 Multiplying Radical Epressions. Use the rules we have discovered to simplify these more difficult epressions. Eamples: Simplify Practice: Simplify Working with variables: Don t be tricked by these easy ones! Eamples: Simplify y y 6 Practice: Simplify
4 Squares and Square Roots Algebra 11.1 Practice: Simplify completely. All answers should be left in radical form. DO NOT USE A CALCULATOR ,900 Name Period
5 Squares and Square Roots Algebra 11.1 Practice: Simplify completely. All answers should be left in radical form. DO NOT USE A CALCULATOR y 14. a b y y y y y.
6 Rationalizing the Denominator Algebra 11.4 Simplified Radicals must NOT have a radical in the denominator. Removing the radical is called Rationalizing the Denominator. Eamples: Simplify Practice: Simplify y 4. 4 Combining Like Radicals Eamples: Simplify Practice: Simplify Harder Practice: Simplify
7 Distribution and FOIL Algebra 11.4 Use Distribution and FOIL with radicals just as you would with integers. Distribution Eamples: Simplify Practice: Simplify FOIL Eamples: Simplify Practice: Simplify FOIL Eamples: Perfect Squares and Difference of Squares. 1. ( )( ). ( 6 ) FOIL Practice: Perfect Squares and Difference of Squares. 1. ( 8). ( 6 )( 6 )
8 Distribution and FOIL Algebra 11.4 Review: Rationalize each denominator. Name Period To rationalize a comple radical denominator, multiply by the CONJUGATE. The conjugate is the epression which makes the denominator a difference of squares. Eamples: Rationalize each denominator Practice: Rationalize each denominator
9 Radicals Quiz Review Algebra 11.4 Simplify: Rationalizing the Denominator: Simplify and Rationalize each denominator FOIL and Distribution: Simplify each and rationalize all denominators ( 6 1) 00. ( 1 )( )
10 Name Period Radicals Practice Quiz Algebra 11.4 Simplify each: Answers should be in simplest radical form. Rationalize all denominators. CALCULATORS WILL NOT BE ALLOWED ON THIS QUIZ , y 4.. 8a
11 Name Period Radicals Practice Quiz Algebra 11.4 Simplify each: Answers should be in simplest radical form. Rationalize all denominators. CALCULATORS WILL NOT BE ALLOWED ON THIS QUIZ ( )( 6) ( 1)( 1)
12 Squares and Square Roots Algebra 11.1 Practice: Simplify completely. All answers should be left in radical form. DO NOT USE A CALCULATOR Name Period
13 Name Period Squares and Square Roots Algebra 11.1 Practice: Simplify completely. All answers should be left in radical form. DO NOT USE A CALCULATOR y. y y y a a a y y ( 4 )( 4 )
14 Squares and Square Roots Algebra 11.1 Practice: Simplify completely. All answers should be left in radical form. DO NOT USE A CALCULATOR Name Period
15 Name Period Squares and Square Roots Algebra 11.1 Practice: Simplify completely. All answers should be left in radical form. DO NOT USE A CALCULATOR a. y y a b y 4. 4 y ( )( )
16 The Pythagorean Theorem Algebra 11.4 The Pythagorean Theorem: The sum of the squares of the legs of a right triangle is equal to the square of its hypotenuse. a Simply: a b c Where a and b are the legs and c is the hypotenuse. b The hypotenuse is the longest side, always opposite the right angle. c Eamples: Find the missing length Practice: Find the missing length Practice: Find the missing length. LEAVE ANSWERS IN RADICAL FORM
17 The Pythagorean Theorem Algebra 11. Many word problems can be solved using the Pythagorean Theorem. Eamples: Solve each using the Pythagorean Theorem. 1. A rectangle has a diagonal length of cm and a width of cm. Find its area (leave in simplified radical form).. The wire supporting a 0-foot tall phone pole is attached to the top of the pole, and to the ground 1 feet from the pole. How long is the wire? Practice: Solve each using the Pythagorean Theorem. 1. Find the hypotenuse of a right triangle whose legs are and 4 inches long.. What is the length of the diagonal of a square that has -inch sides? (Leave in simplified radical form.). If you walk 1 mile north, then miles east, then three miles north, how far are will you be from where you started? Pythagorean Triples: You can determine whether a triangle is a right triangle by testing the sides using the Pythagorean Theorem. Eamples: Which of the lengths below could be the sides of a right triangle? Practice: Which of the lengths below could be the sides of a right triangle? Practice: Which triangle is a right triangle?
18 Distance on the Plane Algebra You can use the Pythagorean Theorem to find the distance between two points on the coordinate plane. 11. Practice: Find the length of each segment on the coordinate plane below: AB = (-,)D A (,) B (6,1) BC = CD = AD = C (-6,-4) Of course, the distance between two points on the plane can be found without graphing: Eample: Find the distance between the points (11,-) and (,-11) on the plane. (,-11) (11,-) Given any two points: ( 1, y 1 ) and (, y ): The distance between two points on the plane is the hypotenuse of a right triangle with a width of and a height of. The distance formula IS based on the pythagorean theorem: d ( ) ( y y ) 1 1 d ( ) ( y y ) 1 1
19 Distance on the Plane Practice: Find the distance between each pair of points below: Algebra 11. A (-4,1) B (-8,-) C (8,-4) A(-4,1) to B(-8, -) = A(-4, 1) to C(8, -4) = A(-4, 1) to D(, -) = C(8, -4) to D(, -) = (leave in radical form) D (, -) Midpoint: Try to find the midpoint of each segment below. Look for a relationship that would help you find the midpoint without graphing. (-1,)C A (,) midpoint of AB = midpoint of BC = midpoint of AC = B (, -) Try to write the midpoint formula on your own. Practice: What is the midpoint fr segment MN for M(, -1) and N(, -)
20 Pythagorean Problems (Easier) Practice: Solve each using the Pythagorean Theorem: Name Period Algebra 1. Chase rides his bicycle miles south, then miles west, then 1 mile north. How far is he from where he started (to the tenth of a mile)? 11.. An equilateral triangle has 4-inch sides. What is its height (in simplest radical form)?. Triangle ABC is inscribed (drawn within) in the prism on the right. What is the perimeter of triangle ABC (in simplest radical form)? 8in A 6in 6in B C 4. An ant is crawling along the outside of the bo below. How far will he walk from A to B along the path shown (think about unfolding the bo to solve this problem). A 1in 8in in B. A 41-foot ladder rests against a wall so that the top of the ladder is 40 feet from the ground. How far from the wall is the bottom of the ladder? 6. Solve for : -1
21 Pythagorean Problems (Harder) Practice: Solve each using the Pythagorean Theorem: Name Period Algebra Mary and Benjamin are driving to their friend Paul s house for a birthday party. Mary drives 9 miles north and 6 miles east to get there, while Benjamin drives miles south and miles west. How far does Mary live from Benjamin (round to the tenth of a mile)?. An equilateral triangle has 10-inch sides. What is its area (in simplest radical form)? hint: area of a triangle = bh/. What is the distance from A to B in the following prism (in simplest radical form)? 8in B A 6in 6in 4. An ant is crawling inside of a bo with the dimensions below. What is the shortest possible distance the ant can walk along the inside surface of the bo to get from corner A to the food at corner B? A 8in 6in 0in B. A 0-foot ladder rests against a wall so that the top of the ladder is 48 feet from the ground. As you start to climb the ladder, it slips and the top of the ladder drops 8 feet. How far does the bottom of the ladder slide away from the wall (from its original position)? 6. Solve for : +1-1
22 Pythagorean Review Solve: Name Period Algebra The hypotenuse of a right triangle is 4cm, and one of its legs is 6cm. Find its perimeter. 1.. A fifteen-foot ladder reaches the top of a 1-foot wall. How far is the base of the ladder from the base of the wall? (leave in radical form).. If you drive miles west, then miles south, and finally 1 miles east, how far will you end up from where you started?. 4. An isosceles triangle has two congruent 11-inch sides, and an 18-inch base. What is its area (in simplest radical form)? 4.. Addison is standing in the middle of a large field throwing baseballs. He throws the first ball 0 yards straight out. He turns 90 degrees to the right an throws a second ball yards straight out. He turns 90 degrees to the right again and throws a third ball 4 yards (straight out again). What is the shortest distance he can walk to retrieve all three balls (he does not need to return to his original spot). Round to the tenth..
23 Pythagorean Review Answer each: Name Period Algebra Solve for : + 6. (leave as a simplified fraction). A cube has two-inch edges. What is the distance between opposite corners A and B of the cube? (leave in radical form) B A. 8. An equilateral triangle has 8-inch sides. What is the height of the triangle? (leave in radical form) The short leg of a right triangle is inches long, and the hypotenuse of the triangle is inches. How long is the longer leg (in terms of, leave in radical form)? 9. Challenge. A rectangle is nine inches longer than it is wide, and its diagonal is 10 inches longer than its width. What is the width of the quadrilateral? (Round to the hundredth, or, even better... leave in radical form). C.
24 Pythagorean Review Right Triangles are EVERYWHERE! Prisms: Algebra 11. 4in D in C in A B Name the right triangles you can find in this figure (using only A, B, C and D): Now find the distance from: A to C, B to D, and A to D. Non-Right Triangles: You can use the Pythagorean Theorem to find the altitude (height) of triangles. Practice: Find each height. 1. (equilateral). (Isosceles). (scalene... much harder!) in in in in in 8in 9in More work with variables: The Pythagorean Theorem works even without numbers. Practice: Solve for in each. 1.. (leave as a fraction). (solve as a quadratic)
25 Name Period Pythagorean Practice Test Algebra 11.4 Find the missing length for each diagram below. Leave all irrational answers in radical form = 0.. =. 10. = = 1. (rectangle). = = (leave as a simplified fraction)
26 Name Period Pythagorean Practice Test Algebra 11.4 Solve each. Leave answers in simplest radical form unless noted otherwise.. What is the area of an equilateral triangle with 6-inch sides (leave answer in simplest radical form).. 8. Alonzo walks 40 meters north, then 18 meters east, then 16 meters south, then directly back to where he started. How far did he walk altogether? (Round to the tenth of a meter.) The wires that support a 90-foot antenna are 9 feet long. How far from the base of the tower are the wires attached? (in simplest radical form) Find the midpoint and distance between the following pair of points (to the nearest tenth): (9, -) (-1, ) Midpoint: 10. Distance: (in simplest radical form) What is the diagonal length of two adjoining squares whose side length is cm (in simplest radical form)? What is the diagonal length of a cube whose edge length is cm (in simplest radical form)? The numbers 1 and are part of a Pythagorean Triple. What is the third number in the Pythagorean Triple which includes 1 and? 14.
27 Name Period Pythagorean Practice Test (4th) Algebra 11.4 Solve each. Leave answers in simplest radical form unless noted otherwise.. What is the area of an equilateral triangle with 6-inch sides (leave answer in simplest radical form).. 8. Alonzo walks 40 meters north, then 18 meters east, then 16 meters south, then directly back to where he started. How far did he walk altogether? (Round to the tenth of a meter.) The wires that support a 90-foot antenna are 9 feet long. How far from the base of the tower are the wires attached? (in simplest radical form) Find the midpoint and distance between the following pair of points (to the nearest tenth): (9, -) (-1, ) Midpoint: 10. Distance: (in simplest radical form) What is the diagonal length of two adjoining squares whose side length is cm (in simplest radical form)? Simplify each: a
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