Suggested Approach Pythagorean Theorem The Converse of Pythagorean Theorem Applications of Pythagoras Theorem. Notes on Teaching 3

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Colin Hampton
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1 hapter 10 Pythagorean Theorem Suggested pproach Students can explore Pythagorean Theorem using the GSP activity in lass ctivity 1. There are over 300 proofs of Pythagorean Theorem. Teachers may illustrate some proofs, or ask students to think and search for different proofs. The emphasis of this chapter is on its application to solve problems reducible to right-angled triangles. Students should understand the relationship between Pythagorean Theorem and its converse, and have ample practice on application problems Pythagorean Theorem Students should be able to identify the right angle in a right-angled triangle before applying Pythagorean Theorem. The simple Pythagorean Triples such as (3-4-5, ) should be introduced. Refer to Question 1 of Exercise The onverse of Pythagorean Theorem Teachers may first introduce the idea of the converse of a theorem. Students should understand that the converse of Pythagorean Theorem is a tool used to prove that a triangle is a right-angled triangle pplications of Pythagoras Theorem Pythagorean Theorem can be applied to find the length of a side of a right-angled triangle when two other sides are given. In practical problems, students should identify the right angles first. Sometimes, we need to construct additional lines to form right-angled triangles. esides arousing and increasing students interest in the topic with proofs of the Pythagorean Theorem, students should also be encouraged to explore the role of the theorem in the spiral of Theodorus. Notes on Teaching 3

2 hapter 10 Pythagorean Theorem lass ctivity 1 Objective: To explore the relationship between the sides of a right-angled triangle. Pythagorean Theorem a =.6 cm b = 4.95 cm c = 5.60 cm a = 6.87 cm b = 4.48 cm c = cm c a b c a b c a b.75 cm 4.66 cm 5.41 cm 7.56 cm 1.68 cm 9.4 cm.09 cm 3.55 cm 4.1 cm 4.38 cm 1.57 cm cm.81 cm 3.55 cm 4.5 cm 7.88 cm 1.57 cm 0.45 cm.6 cm 4.95 cm 5.60 cm 6.87 cm 4.48 cm cm.6 cm 4.95 cm 5.60 cm 6.87 cm 4.48 cm cm Tasks (a) Draw a right-angled with = 90. (Hint: Use the command onstruct Perpendicular line to draw a line passing through and perpendicular to.) Label the sides opposite,, and as a, b, and c respectively. (c) Measure the lengths of a, b, and c. (d) Find the squares of a, b, and c. (e) Select all the measurements in steps (c) and (d). Then select the command Graph Tabulate to create a table of values of a, b, c, a, b, and c. (f ) Drag the vertices of around to obtain another set of values of the sides and their squares. Double-click the table to add the current measurements as a new row to it. (g) Repeat step (f ) for two or more sets of measurements. Question What is the relationship between a, b, and c in the right-angled? a + b = c hapter 10 Pythagorean Theorem 57

3 lass ctivity Objective: To explore one of the proofs of the Pythagorean Theorem. Tasks (a) triangle with vertices O(0, 0), (a, 0) and (0, b), and a square with vertices O(0, 0), P(a + b, 0), Q(a + b, a + b), and R(0, a + b), where a and b are real numbers, are drawn on the artesian plane in the diagram. a + b y R (a + b, a + b) Q a D b O b a P a + b x On RQ, mark the point with coordinates (b, a + b) and on PQ, mark the point D with coordinates (a + b, a). Join the points,,, and D to obtain figure D. (c) Why is the figure D a square? Explain your answer. ll sides of figure D are hypotenuses of triangles with the same dimensions as no. (d) Giving your answers in terms of a and b, find (i) the area of the square OPQR, rea of OPQR = base 3 height = (a + b ) 3 (a + b ) = (a + b ) (ii) the area of each triangle. rea of each triangle = 1 3 base 3 height = 1 3 a 3 b = 1 ab 58

4 (e) Suppose the length of is c units, find an expression in terms of a, b, and c for the area of D. = O + O (Pythagorean Theorem) = a + b \ = a + b If = c units, c = a + b rea of D = 3 D = a + b 3 a + b = ( a + b ) = a + b = c (f ) Relating your expression in (e) with O, what conclusion can you draw? rea of D = rea of OPQR rea of 4 triangles = (a + b ) 4 1 ab = a + ab + b ab = a + b \ a + b = c lass ctivity 3 Objective: To apply Pythagorean Theorem to construct lengths of irrational square-root values on the number line diagram. Tasks (a) On a sheet of graph paper and taking cm to represent 1 unit, draw a unit square O 1 1 as shown below. Extend the lines O 1 and 1 to OX and 1 respectively. Line OX is considered as a number line O X hapter 10 Pythagorean Theorem 59

5 hapter 10 Pythagorean Theorem 61 Extend Your Learning urve Diagonal of a Prism Research and write a brief report on this technology and its possible applications. rectangular prism is a 3-dimensional figure with a rectangular base and top of the same size which are parallel to each other, and each of the other sides is also a rectangle. rectangular prism with edge lengths, l, w, and h is shown below. h l w (a) Find the length of the diagonal which is drawn from one corner of the base to the opposite corner of the top, as shown in the figure, in terms of l, w, and h. What is the length of the diagonal joining the opposite corners of a prism whose base is 3 cm by 4 cm and height is 1 cm? Suggested nswers: (a) (length of diagonal of base) = l + w (Pythagorean Theorem) length of diagonal of base = ( l + w ) units length of diagonal from base to top = ( l + w ) + h = l + w + h units Let h = 1 cm, w = 3 cm and l = 4 cm. length of diagonal = l + w + h = = 169 = 13 cm

6 Exercise 10.1 In this exercise, give your answers rounded to 3 significant figures where necessary. asic Practice 1. Find the unknown side of each of the following triangles, given that the measurements are in centimeters. (a) F 6 b 8 D e 0 E 1 3. PQR has a right angle at R with PQ = 61 cm and QR = 11 cm. What is the length of PR? P PR = = 3,600 PR = 3, 600 = 60 cm 61 (Pythagorean Theorem) Q 11 R (c) 1 G h (d) 4 P q 4. In XYZ, X = 90, YZ = 7 ft, and XY = 3 ft. Find the length of XZ. H 37 K Q 6 R Y (a) b = (Pythagorean Theorem) = 100 b = 100 = 10 cm e = (Pythagorean Theorem) = 841 e = 841 = 9 cm (c) 37 = 1 + h (Pythagorean Theorem) h = 1,5 h = 1, 5 = 35 cm (d) 6 = 4 + q (Pythagorean Theorem) q = 100 q = 100 = 10 cm. In, = 90, = 9 in., and = 40 in. What is the length of? = = 1,681 = 1, 681 = 41 in. 40 (Pythagorean Theorem) 9 Z 7 = 3 + XZ (Pythagorean Theorem) XZ = 40 XZ = 40 = 6.3 ft (rounded to 3 sig. fig.) Further Practice 5. In the figure, D = D = 90, = 3 cm, D = 1 cm, and D = cm. Find the length of (a) D,. D 1 (a) In nd, D = (Pythagorean Theorem) = 10 D = 10 = 3.16 cm (rounded to 3 sig. fig.) 3 In nd, = D + D (Pythagorean Theorem) = + 10 = 14 = 3.74 cm (rounded to 3 sig. fig.) 7 3 X 64

Time : 2 Hours (Pages 3) Max. Marks : 40. Q.1. Solve the following : (Any 5) 5 In PQR, m Q = 90º, m P = 30º, m R = 60º. If PR = 8 cm, find QR.

Time : 2 Hours (Pages 3) Max. Marks : 40. Q.1. Solve the following : (Any 5) 5 In PQR, m Q = 90º, m P = 30º, m R = 60º. If PR = 8 cm, find QR. Q.P. SET CODE Q.1. Solve the following : (ny 5) 5 (i) (ii) In PQR, m Q 90º, m P 0º, m R 60º. If PR 8 cm, find QR. O is the centre of the circle. If m C 80º, the find m (arc C) and m (arc C). Seat No. 01