G8 Square Roots and Pythagoras Theorem G8.1 Square and Square Roots Definitions: If a a = n, then (1) n is the square of a, i.e. n = a. () a is the square root of n. For example, since = 4 and ( ) ( ) = 4, (1) The square of is 4 and the square of is also 4. () and are the square roots of 4. The symbol for square root is, called the radical sign. We may write 4 =. In fact, for any positive number n, there are two square roots, the positive square root n and the negative square root n. So if a = 9, then a = 9 = 3 or a = 9 = 3. Note that 9 = 3 but 9 3. Checkpoint 1 By the use of a calculator, evaluate the following: (Give the answers correct to 1 decimal place where necessary.) (a) ( 9) (b) ( 1.1) (c) 361 (d) 89 (e) 0.05 + 1. 64 (f) (g) 3 1 3 + 1 (h) 6 + 3 3 3 5 1
Facts: For positive numbers a and b, (1) a b = a b ; a a () =. b b We may use these results to evaluate square roots and expressions involving square roots. Example 1 Evaluate the following without using a calculator: (a) 484 (b) 1764 (c) 1600 (d) 0. 04 (e) 36 49 (f) 1 4 Solution (a) 484 = 11 11 = = 11 = 11 = 11 (b) 1764 = 3 3 7 7 = 3 7 = 3 = 3 7 = 4 7 (c) 1600 = 16 100 = 16 100 = 4 10 = 40 (d) 0.04 = 4 0.01 = 4 0.01 = 0.1 = 0. (e) (f) 36 36 6 = = 49 49 7 1 9 9 = = = 4 4 4 3
Checkpoint Evaluate the following without using a calculator: (a) 1369 (b) 6400 (c). 56 (d) 15 9 (e) 34 1156 (f) 76 4 81 3
G8. Pythagoras Theorem The figure shows a right-angled triangle ABC. The longest side, which is opposite to the right angle, is called the hypotenuse. Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. i.e. If ABC is right-angled at C, then [Reference: Pyth. Theorem] a = + b c. Proof: Consider two congruent right-angled triangles ABC and PAQ placed as shown in the figure so that QAC is a straight line. We are going to find the area of the trapezium BCQP in two different ways. (1) Area of trapezium BCQP = Area of ABC + Area of 1 1 1 = ab + ab + c 1 = (ab + c ) PAQ + Area of PBA () Area of trapezium BCQP 1 = (BC + PQ) QC 1 = ( a + b)( a + b) 1 = ( a + ab + b ) i.e. 1 1 (ab + c ) = ( a + ab + b ) ab + c = a + ab + b a + b = c 4
Example In the figure, find the value of b. Solution 9 + ( 40) b = b = 81+ 40 = 11 b = 11 = 11 Example 3 In the figure, find the values of x and y. (Give the answers correct to decimal places where necessary.) Solution (a) In ABC, x = 6 + 8 = 100 x = 100 = 10 (b) In ACD, 4 + y 16 + y y = x = 100 = 100 16 = 84 y = 84 = 9.17 (corr. to decimal places) 5
Checkpoint 3 In each of the following figures, find the values of the unknowns. (Correct the answers to decimal places where necessary.) (a) (b) (c) 6
Example 4 In the figure, B = 90 o. BC = CD = DE = EF. Prove that 7AC + 3AF = 7AD + 3AE Solution In ABC, AC + = AB BC In ABD, AD = + AB = AB = AB BD + (BC) + 4BC In ABE, AE = + AB = AB = AB BE + (3BC) + 9BC In ABF, AF = + AB = AB = AB BF + (4BC) + 16BC 7AC = 7(AB = 10AB + 3AF + BC ) + 3(AB + 55BC + 16BC ) 7AD = 7(AB = 10AB + 3AE + 4BC ) + 3(AB + 55BC + 9BC ) 7AC + 3AF = 7AD + 3AE 7
G8.3 Applications of Pythagoras Theorem There are many problems that involve right-angled triangles and some times Pythagoras Theorem can be used to solve them. Example 5 A ladder 4 m long leans against a vertical wall. Its foot is 1. m from the wall. How far up the wall will the ladder reach? (Correct the answers to decimal places.) Solution Let h m be the height reached. h h + (1.) = 4 + 1.44 = 16 h = 16 1.44 = 14.56 h = 14.56 = 3.8 (corr. to decimal places) The ladder will reach a height of 3.8 m. Example 6 A boat sails 6 km due north and then 4 km due east. How far is it from its starting point? (Correct the answers to decimal places.) Solution Let d km be the required distance. d d d = 6 + 4 = 5 = 5 = 7.1 (corr. to decimal places) The boat is 7.1 km from its starting point. 8
Checkpoint 4 Ship A and Ship B leave a port together. Ship A sails due east at 15 km/h and Ship B sails due north at 0 km/h. What is the distance between Ship A and Ship B after 3 hours? Checkpoint 5 Two vertical walls are 4 m apart. One post is m high and the other is 15 m high. What is the distance (d) between the tops of the two posts? 9
G8.4 Converse of Pythagoras Theorem The converse of Pythagoras Theorem is also true. i.e. In ABC, if a = + b c, then C = 90 o. [Reference: Converse of Pyth. Theorem] c A b Proof: B a Construct a right-angled triangle A B C such that C = 90 o, B C = a and A C = b. C By Pythagoras Theorem, we have A b B a C Since We then have C = C = 90 o. (By construction) (By construction) (proved) (SSS) Example 7 In the figure, determine whether ABC is right-angled. Solution AB BC = 13 = 1 = 169 = 144 AC = 5 = 5 Q BC + AC = AB C = 90 (Converse of Pyth. Theorem) Hence ABC is right-angled at C. 10
Checkpoint 6 In the following figures, determine whether they are right-angled triangles. (a) (b) 11
G8.5 Rational Numbers and Irrational Numbers Definitions: m (1) A rational number is a number that can be expressed as a fraction, where m and n are n integers. () An irrational number is a number that is not a rational number, i.e. a number that cannot be expressed as a fraction m, where m and n are integers. n Example 8 1 5,, 1, 3. are rational numbers. 3 5 Proof: 5 m 5 = =, where m = 5 and n = 1; 1 n m =, where m = and n = 3; 3 n 1 6 1 =, where m = 6 and n = 5; 5 5 16 3. =, where m = 16 and n = 5. 5 1 5,, 1, 3. are rational numbers. 3 5 1
Example 9 is an irrational number. Proof: (by Suppose the Converse ) Suppose is a rational number. Then m = or n n = m, where m and n are integers and they do not have common factors except 1. We have m = n (*). Since m and n are integers, is a factor of m. So m is an even number. Also m is an even number and can be expressed as m = k. By equation (*), we have 4k = (k) = m = n or k = n. Since n and k are integers, is a factor of n. Hence n is an even number and so n is also an even number. But m and n have a common factor. This contradicts our supposition. Thus we can conclude that our supposition is a false statement, i.e. is not an irrational number or is a rational number. Usually, radicals like, 6 are irrational numbers. But 4 is a rational number since 4 = =. 1 There are non-radical irrational numbers like π. Checkpoint 7 Are the following numbers rational or irrational? (a) + 5 (b) 3 3 (c) (d) π (e) 3 3 13
G8 Exercises Square Roots and Pythagoras Theorem G8.1 1. Evaluate the following with a calculator: (a) 3 5 (b) 3 3 + (c) 3 + 5 1 (d) 1 + 3. Evaluate the following without using a calculator: (a) 79 (b) 14400 (c) 0. 0005 (d) (e) 46 (f) 49 3 196 56 1 3 G8. 3. Find the values of the unknowns in the following figures. (a) (b) (c) (d) 14
(e) (f) (g) (h) (i) (j) 4. In the figure, ABC is a triangle right-angled at B. BD AC. AB = 6 cm and BC = 8 cm. (a) Find the length of AC. (b) Find the area of ABC. (c) Using the result of (a) and (b), find the length of BD. (d) Find the length of AD. G8.3 5. A boat sails 3 km due east and then 3.5 km due south. How far is the boat now from its starting point? (Correct the answer to 1 decimal place.) 15
6. Two flagpoles are 7 m and 10 m tall respectively. If the distance between the tops of the two flagpoles is 10 m, find the horizontal distance between them. (Give the answers correct to decimal places.) 7. A ladder 13 m long is placed against a vertical wall and reaches a height of 1 m. If the top of the ladder slides 4 m down the wall, how far will the foot of the ladder slide? (Correct the answer to decimal places.) G8.4 8. Determine whether the following triangles are right-angled. (a) (b) (c) (d) 16
9. In the figure, ABC is a triangle with AB = BC = a cm an CA = a cm. Show that B is a right angle. 10. In the figure, XYZ is a triangle. XN is the perpendicular from X to YZ. If YN = 9 cm, NZ = 16 cm and XN = 1 cm, show that YXZ = 90 o. Supplementary 11. In the figure, ABC is a right-angled triangle. D and E are the mid-points of two shorter sides BC and AB respectively. Prove that 4 (AD = + CE ) 5AC. 17