Square Roots and Pythagoras Theorem

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1 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) (f) (g) (h)

2 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) (e) (f) 1 4 Solution (a) 484 = = = 11 = 11 = 11 (b) 1764 = = 3 7 = 3 = 3 7 = 4 7 (c) 1600 = = = 4 10 = 40 (d) 0.04 = = = 0.1 = 0. (e) (f) = = = = =

3 Checkpoint Evaluate the following without using a calculator: (a) 1369 (b) 6400 (c). 56 (d) 15 9 (e) (f)

4 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 = 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

5 Example In the figure, find the value of b. Solution 9 + ( 40) b = b = = 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 = = 100 x = 100 = 10 (b) In ACD, 4 + y 16 + y y = x = 100 = = 84 y = 84 = 9.17 (corr. to decimal places) 5

6 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

7 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.

8 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.) = = 16 h = = h = = 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 = = 5 = 5 = 7.1 (corr. to decimal places) The boat is 7.1 km from its starting point. 8

9 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

10 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

11 Checkpoint 6 In the following figures, determine whether they are right-angled triangles. (a) (b) 11

12 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 =, where m = 6 and n = 5; =, where m = 16 and n = ,, 1, 3. are rational numbers

13 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)

14 G8 Exercises Square Roots and Pythagoras Theorem G Evaluate the following with a calculator: (a) 3 5 (b) (c) (d) Evaluate the following without using a calculator: (a) 79 (b) (c) (d) (e) 46 (f) G8. 3. Find the values of the unknowns in the following figures. (a) (b) (c) (d) 14

15 (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. G 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

16 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.) G Determine whether the following triangles are right-angled. (a) (b) (c) (d) 16

17 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

CHAPTER 10 TRIGONOMETRY

CHAPTER 10 TRIGONOMETRY EXERCISE 39, Page 87 1. Find the length of side x in the diagram below. By Pythagoras, from which, 2 25 x 7 2 x 25 7 and x = 25 7 = 24 m 2. Find the length of side x in the diagram