**SUBJECT: MATHEMATICS**

**CLASS: JSS 2**

**DATE:**

**TERM: 3rd TERM**

**REFERENCE**

- WABP ESSENTIAL MATHEMATICS FOR JSS BK 2 BY A.J.S. OLUWASANMI
- NEW GENERAL MATHEMATICS BY J.B. CHANNON & ETAL

WEEK NINE

**TOPIC: PYTHAGORAS THEOREM (SOLUTION OF TRIANGLE)**

**CONTENT:**** **i. Pythagoras triple

- Pythagoras theorem

iii. Using Pythagoras theorem to solve other related problems.

**PYTHAGORAS TRIPLE**

The sides of a right-angled triangle can be related to the proof of Pythagoras Triple. A Pythagoras triple is a set of three whole numbers which numbers which gives lengths of the sides of right-angled triangle.

Examples of some common Pythagoras triple are (3, 4, 5), (6, 8, 10). (5, 12, 13), etc.

**Worked Example**

Which of the following is a Pythagoras triple?

- a) (15, 30, 35) b) (33, 56, 65)

**Solution **

152 + 302 = 225 + 900

= 1125

But 352 = 1225

(15, 30, 35) is not a Pythagoras triple

- b) 332 + 562 = 1089 + 3136 = 4225

652 = 4225

Thus, 332 + 562 = 652

(33, 56, 65) is a Pythagoras triple.

**Evaluation: Class Work**

Find out which of the following are Pythagoras triples.

- a) (12, 16, 20) b) (27, 36, 45) c) (14, 24, 28)

**Answer to the evaluation question**

- (12, 16, 20)

122 + 162 = 144 + 256 = 400

202 = 400

Thus, 122 + 162 = 20

(12, 16, 29) is a Pythagoras triple.

- (27, 36, 45)

272 + 362 = 729 + 1296 = 2025

452 = 2025

Thus, 272 + 362 = 452

(27, 36, 45) is a Pythagoras triple

- (14, 24, 28)

142 + 242 = 196 + 576 = 772

282 772

Thus, 14, 24, 28 is not a Pythagoras triple.

**Reference: **New General Mathematics Book 2, Chapter 7, Pages 150 – 151

Essential Mathematics for JSS Book 2, Chapter, 21, pages 218 and 219

**PYTHAGORAS THEOREM**

The Pythagoras’ Theorem states that in any right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the two sides.

/AB/ = hypotenuse, /BC/ and /AC/ are the other two sides, i.e.

/AB/2 = /BC/2 + /AC/2

Since /AB/ = c, /AC/ = b, /BC/ = a

Then, c2 = a2 + b2

**Worked Examples**

Calculate the length of the two sides of each of the triangle below

- a) b)

**Solution **

- Using Pythagoras rule

C2 = a2 + b2

a = 3, b = 4

c2 = 32 + 42 = 9 + 16

c2 = 25

c = 5m, the length of the third side is 5m.

- Using Pythagoras rule

c2 = a2 + b2

C = 13, a = a b = 5

132 = a2 + 52

a2 = 169 – 25 = 144

a = 144

a = 12cm

find the length of the third side of the triangle below:

- a) b) c)

**Answer to the evaluations.**

- /AC/2 = /AB/2 + /BC/2

AC = ?, AB = 8cm, BC = 6cm

AC2 = 82 + 62

AC = 100 = 10cm

- /AC/2 = /AB/2- /BC/2

AC = 100m, AB = 80m, BC =?

1002 = 802 + /BC/2

1000 = 6400 + /BC/2 ∴ /BC/2 = 1000 – 6400

/BC/ = 3600 = 60M

- /AC/2 = /AB/2 + 72

AC = 25, /AB/ = 72

252 = /AB/2 + 49

/AB/2 = 625 – 49 = 576

/AB/ = 576 = 24cm

**Reference **

NGM BK 2, chapter 17, pages 147 – 148

Essential mathematics for JSS BK 2, chapter 21, pages 215 – 218

**USING PYTHAGORAS THEOREM TO SOLVE OTHER RELATED PROBLEM INVOLVING TRIANGLES**

In some cases, we may have more than one right – angled triangle.

**Worked examples**

- Calculate the length of the unknown in the following triangle:

**Solution**

- PRS is right angled triangle, PQR is also a right angled triangle

Let PR beycm

In triangle PQR; y2 = 32 + 22

= 9 + 4 = 13

∴ y2 = 13

Let PS be xcm

In triangle PRS, x2 = y2 + 62

Substitute 13 for y2 in the formula

x2 = 13 + 62

x2 = 13 + 36

x2= 49 = 7

PS = 7cm

- AD is the right angled ABD. Let AB be ycm.

In triangle ABC, x2 = y2 + (8 + 12)2

Substitute 225 for y2 in the formula

X2 = 225 + 202

= 225 + 400 = 625

X = 635 = 25cm

Therefore, AD = 25cm

When solving triangle relating to decimal fraction and whole numbers, it is advisable to find the squares and square root from tables or multiplying the decimal by itself.

**Evaluation**

- A ladder is 7.3m long and the foot of the ladder is 1.8m from the wall. How far up the wall is the ladder?
- The distances between the opposite corner of a rectangular lawn is 30m, of the lawn is 24m. Calculate the breadth of the lawn.

**GENERAL EVALUATION**

- The distance between the opposite corners of a rectangular plot is 30m. The length of the plot is 24m. Calculate the breadth of the plot.
- A student cycles from home to school, first eastwards to a road junction 12km from home, then southwards to school. If the school is 19km from home, how far is it from the road junction?

**REVISION QUESTION:**

- A square top lid of a container has a diagonal 150cm. Find the length of one side of the lid.
- ABCD is a rectangle. AB = xcm, BC = 9cm and the diagonal AC = 19cm. Calculate the value of x.

**READING ASSIGNMENT**

Essential Mathematics for JSS 2 Chapter 21 pages 268 – 271

Exercise 21.1 1a – b, 2a – d, 3a – b, page 270

**WEEKEND ASSIGNMENT**

- The longest side of a right-angled triangle is called A. hypotenuse C. hypostasis C. base D. adjacent
- Calculate the length of the diagonal of a room 15m by 12m. A. 9m B. 81m C. 19m D. 12m
- Which of the following are Pythagorean triples? A. 6, 8. 10 B. 12, 28, 32 C. 9, 12, 20 D. 13, 15, 17
- Calculate the value of x in the diagram below.

- 25m B. 15m C. 5m D. 11m

- In the diagram below, which of the following gives the value of side x2?

- x2 = z2 + y2 B. x2 = z2 – y2 C. x2 = y2 – z2 D. x = z2 – y2

**THEORY**

- A flagpole 5m tall is supported by a wire that is fixe at point 3m from the base of the pole. Calculate to 1 d.p the length of the wire.
- A square top lid of a container has a diagonal of 150cm. Find the length of one side of the lid.

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