i “Handout-15” — 2019/2/2 — 17:24 — page 1 — #1 i i i Math 5: Handout 15 MATH 5: ASSIGNMENT 5 Square Root. Pythagorean Theorem. FEB 21, 2010 Square roots • Square root of a is a number whose square is equal to a. For example: square root of 25Material is 5, because covered52 = 25. today p • Notation: square root of number a is commonly denoted a. p p p p p Recall:p Square root of a (denoted √a)isanumberwhosesquareisequaltoa.Forexample:squareroot • ab = a b, but a + b is not equal to aof+ 25b is. 5, because 52 =25. • Square roots naturally appear in geometry: We discussed today that √ab = √a√b,but√a + b is not equal to √a + √b. Square roots naturally appear in geometry: Pythagorean theorem: In a right triangleTheorem with legs a, b(Pythagoreanand hypotenuse theorem)c, one has. In a right triangle with legs a, b and hypotenuse c,onehas a2 + b2 = c2 a2 + b2 = c2 p or c = a2 + b2 or c = √a2 + b2 Proof: Consider the following picture: Aproofofthistheoremisillustratedbelow: In this square, the total area is In this square, the total area is 2 2 2 2 (a + b) 2 (a + b)=a 2 (a +2 b)+b (2a + b)=a + ab + ab + b = a +2ab + b ¹a + bº × ¹a + bº = a × ¹a + bº + b × ¹a + bº = a× + ab + ab + ×b = a + 2ab +×b On the other hand, the area of each triangle is 1 ab,andtheareaofshadedsquareisc2.Thus,weget On the other hand, the area of each triangle is 1 ab, and the area of shaded square is c2. Thus, we2 get 2 2 2 1 2 a +2ab + b =4 2 ab + c a2 + 2ab + b2 = 4 × 1 ab + c2 × which gives a2 + b2 =2 c2. 2 2 2 √ which gives a + b = c . For example, in a square with side 1, the diagonal has length 2. It is possible —p but not easy — to find a right triangle where all sides are whole numbers. The easiest For example, in a square with side 1, the diagonalsuch triangle has length is the2 triangle. with sides 3, 4, 5. It is possible — but not easy — to find a right triangle where all sides are whole numbers. The easiest such triangle is Homework the triangle with sides 3, 4, 5. For the homework, let me remind you that you should write solutions, showing your calculations and your 1 Power 2 reasoning — not just answers! Do not write on this homework assignment — use a separate sheet of paper instead; leave the homework assignment in your folder for future reference. We know how to raise numbers into whole powers: 0. Do the practice Math Kangaroo olympiad (attached). n 1. Cana = youa × · find · · × aa. right triangle where all sides are whole numbers and the hypotenuse is 13? 1 But what is a 2 ? 2. If, in a right triangle, one leg has length 1 and the hypotenusehaslength2,whatistheotherleg? 1 Example: Let’s try to figure out what 4 2 is: √ 6 2 1 4 3. Find 2 7 (hint: you do not need to compute what this number itself is...); 16 ; 9 ; 1 1 1 + 1× 1 4 2 × 4 2 = 4 2 2 = 4 = 4. 4. Find the height and area of the figure below. Lengths of three sides are give; the! two marked! angles 1 We can see that 4 2 must be a number, such that if weare multiply right angles. it by itself, we get 4. But this is just a square root of 4! So, we get: 1 p 4 2 = 4. In general, this is also true: 1 p a 2 = a. *5. Remember the grid for writing secret messages we discussed today? Try constructing one like this (using 8 8square).Foryourconvenience,usethetemplateonthenextpage. × i i i i MATH 5: ASSIGNMENT 5 FEB 21, 2010 Material covered today Recall: Square root of a (denoted √a)isanumberwhosesquareisequaltoa.Forexample:squareroot of 25 is 5, because 52 =25. We discussed today that √ab = √a√b,but√a + b is not equal to √a + √b. Square roots naturally appear in geometry: Theorem (Pythagorean theorem). In a right triangle with legs a, b and hypotenuse c,onehas a2 + b2 = c2 or c = √a2 + b2 Aproofofthistheoremisillustratedbelow: In this square, the total area is (a + b) (a + b)=a (a + b)+b (a + b)=a2 + ab + ab + b2 = a2 +2ab + b2 × × × 1 2 On the other hand, the area of each triangle is 2 ab,andtheareaofshadedsquareisc .Thus,weget a2 +2ab + b2 =4 1 ab + c2 i “Handout-15” — 2019/2/2 — 17:24 — page 2 — #2 × 2 i which gives a2 + b2 = c2. √ i For example, in a square with side 1, the diagonal has length 2. i It is possible — but not easy — to find a right triangle where all sides are whole numbers. The easiest Homework such triangle is the triangle with sides 3, 4, 5. 1. Find the following square roots. If you can not find the number exactly, at least say betweenHomework which two whole numbers the answer is, e.g., between 5 and 6. For the homework, let me remind you that you should write solutions, showing your calculations and your p p reasoning —p not just answers! Do notp write on this homeworkp assignment — use a separate sheet of paper a. 16 b. 81 instead; leavec. 10 the, 000 homework assignmentd. 108 in your folder fore. future50 reference. 0. Do the practice Math Kangaroo olympiad (attached). 2. Can you find a right triangle where all sides are whole numbers and the hypotenuse is 13? 1. Can you find a right triangle where all sides are whole numbers and the hypotenuse is 13? 3. If, in a right triangle, one leg has length 1 and2. theIf, hypotenuse in a right triangle, has length one 2, leg what has is length the other 1 and leg? the hypotenusehaslength2,whatistheotherleg? p q q 6 2 1 4 6 2 1 4 4. Find 2 × 7 ; 16 ; 9 ; 3. Find √2 7 (hint: you do not need to compute what this number itself is...); ; ; × 16 9 5. Find the height and area of the figure below.4. LengthsFind the of three height sides and are area given; of the the figure two marked below. Lengths angles are of right three angles. sides are give; the! two marked! angles are right angles. *5. Remember the grid for writing secret messages we discussed today? Try constructing one like this 6. The side of an equilateral triangle is 1 m. Find its height and the area. (using 8 8square).Foryourconvenience,usethetemplateonthenextpage. × x 1 7. Take some positive number x < 100 and using calculator (or computer) calculate the number 2 + x . Call the result x and repeat the same calculation with the new x. Do it 10 times. Then take the result and square it. What did you get? Try to do the same thing starting with different numbers. Is it surprising? 2 i i i i.