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Unformatted text preview: S3 Credit Pythagoras Theorem
Investigating Pythagoras Theorem www.mathsrevision.com Finding the length of the smaller side Solving Real Life Problems Pythagoras Theorem Twice Converse of Pythagoras Theorem
13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. Starter Questions
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Q1. Q2. Explain why 4 + 6 x 5 = 34 and not 50 Calculate a 2  b 2 wh e r e a = 4 and b = ( 5 )
Q3. Q4. Does x2 121 factorise to (x 11) (x 11) The cost of an iPod is 80 including VAT. How much is the iPod BEFORE VAT. NONCALCULATOR
13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. Right Angle Triangles
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c b a
Aim of today's Lesson `To investigate the rightangle triangle and to come up with a relationship between the lengths of its two shorter sides and the longest side which is called the hypotenuse. ` 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. Right Angle Triangles
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What is the length of a ? 3 4 c b a
13 Feb 2012 What is the length of b ? Copy the triangle into your jotter and measure the length of c 5 Compiled by Mr. Lafferty Maths Dept. Right Angle Triangles
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What is the length of a ? What is the length of b ? 6 8 c b
Copy the triangle into your jotter and measure the length of c 10
a
13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. Right Angle Triangles
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What is the length of a ? What is the length of b ? 5 12 c b Copy the triangle into your jotter and measure the length of c 13
a
13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. Right Angle Triangles
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Copy the table below and fill in the values that are missing a 3 5 6
13 Feb 2012 b 4 12 8 c 5 13 10 a2 b2 c2
c b a Compiled by Mr. Lafferty Maths Dept. Right Angle Triangles
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Can anyone spot a relationship between a2, b2, c2. a 3 5 6
13 Feb 2012 b 4 12 8 c 5 13 10 a2 9 25 36 b2 16 144 64 c2 25 169 100
c b a Compiled by Mr. Lafferty Maths Dept. a2 +b2 = c 2 Pythagoras's Theorem
www.mathsrevision.com b a c a +b =c
2 2 2 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. Pythagoras's Theorem x
www.mathsrevision.com ( xy ) + ( yz ) = ( xz )
2 2 2 13 Feb 2012 y Compiled by Mr. Lafferty Maths Dept. z Summary of Pythagoras's Theorem www.mathsrevision.com a +b =c
2 2 2 Note: The equation is ONLY valid for right angled triangles. 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. Calculating Hypotenuse
S3 Credit Learning Intention Success Criteria 1. Know the term hypotenuse " the longest side" 2. Use Pythagoras Theorem to calculate the hypotenuse. www.mathsrevision.com 1. Use Pythagoras Theorem to calculate the length of the hypotenuse "the longest side" 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. Calculating Hypotenuse
S3 Credit Two key points when dealing with rightangled triangles
The longest side in a rightangled triangle is called The HYPOTENUSE The HYPOTENUSE is ALWAYS opposite the right angle www.mathsrevision.com b
13 Feb 2012 c a c2 = a2 + b2 (xz)2 = (xy)2 + (yz)2 x y z Compiled by Mr. Lafferty Maths Dept. Calculating the Hypotenuse
Example 1 Q2. Calculate the longest length of the right below. angled triangle www.mathsrevision.com c 2 = a 2 +b 2 c 2 = 12 2 + 8 2
c2 =208 c = 2 0 8 = 14 .4 2 k m
13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. c 12 8 Calculating the Hypotenuse
Example 2
Q1. An aeroplane is preparing to land at Glasgow Airport. It is over Lennoxtown at present which is 15km from the airport. It is at a height of 8km. How far away is the plane from the airport? www.mathsrevision.com (GA)2 = (GL) 2 + ( LA ) 2 (GA)2 = 15 2 + 8 2
(GA) = 2 8 9
2 A
LA = 8 GA = 2 8 9 = 17 k m
13 Feb 2012 G Airport GL = 15 Lennoxtown L Compiled by Mr. Lafferty Maths Dept. Calculating Hypotenuse
S3 Credit www.mathsrevision.com Now try Ex 2.1 and 2.2 MIA Page 147 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. S3 Starter Questions
S3 Credit 1. Does www.mathsrevision.com 4 4 5 + 5 =11 5 9
y(2 y + 3x + 4) 2. 3. 4.
13 Feb 2012 Expand Does 8d 2 + 4d f act or ise t o 4d 2 (2d +1) 3 6 5 25
Compiled by Mr. Lafferty Maths Dept. Length of the smaller side
S3 Credit Learning Intention Success Criteria 1. Use Pythagoras Theorem to find the length of smaller side. 2. Show all working. www.mathsrevision.com 1. To show how Pythagoras Theorem can be used to find the length of the smaller side. 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. Length of the smaller side
S3 Credit www.mathsrevision.com To find the length of the smaller side of a rightangled triangle we simply rearrange Pythagoras Theorem.
Example : Find the length of side a ? Check answer ! Always smaller than hypotenuse c 2 = a 2 +b 2 a2 = c 2 b 2 a 2 = 2 0 2 12 2 a2 = 2 5 6
a = 2 5 6 = 16 c m
Compiled by Mr. Lafferty Maths Dept. 20cm a cm 12cm 13 Feb 2012 Length of the smaller side
S3 Credit www.mathsrevision.com Example : Find the length of side a ? c 2 = a 2 +b 2 b 2 = c 2 a2 b 2 = 10 2 8 2
b2 =36 10cm 8 cm b cm Check answer ! Always smaller than hypotenuse b = 36 =6cm
Compiled by Mr. Lafferty Maths Dept. 13 Feb 2012 Length of smaller side
S3 Credit www.mathsrevision.com Now work through Ex3.1 and Ex 3.2 Odd Numbers Only 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. S3 Starter Questions
S3 Credit 1.
www.mathsrevision.com Calculat e 3 1 16 = =1 4 4 16
2.56 104 2. W r it e in scient if ic not at ion 45 3. 4.
13 Feb 2012 Fact or ise k 2  6k +8 Explain why 65% of 40 = 26
Compiled by Mr. Lafferty Maths Dept. S3 Credit Solving Real Life Problems Using Pythagoras Theorem
Learning Intention Success Criteria 1. Apply Pythagoras Theorem to solve reallife problems. 2. Show all working. www.mathsrevision.com 1. To show how Pythagoras Theorem can be used to solve reallife problems. 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. Solving RealLife Problems
S3 Credit www.mathsrevision.com When coming across a problem involving finding a missing side in a rightangled triangle, you should consider using Pythagoras' Theorem to calculate its length.
Example : A steel rod is used to support a tree which is in danger of falling down. What is the height of the tree ? c 2 = a2 + b2 a = 17  8
2 2 2 a 2 = 225 c 17m rod b a a = 225 = 15m
13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. 8m Solving RealLife Problems
S3 Credit www.mathsrevision.com Example 2 A garden has a fence around its perimeter and along its diagonal as shown below. What is the length of the fence from D to C. ( AC )2 = (DC )2 + ( AD )2 A B 13m b m C (DC )2 = 132  52
(DC )2 = 144 5m D
Compiled by Mr. Lafferty Maths Dept. DC = 144 = 12m
13 Feb 2012 Length of smaller side
S3 Credit www.mathsrevision.com Now work through Ex4.1 and Ex 4.2 Odd Numbers Only 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. S3 Starter Questions
S3 Credit 1.
www.mathsrevision.com 3 1 1 True or f alse 5 + 2 = 7 4 8 8
I f p = 4 , q = 3 and r = 2. Find 2q 2 z + r 2 2. 3.
13 Feb 2012 Fact orise 24 y 16 y 2
Compiled by Mr. Lafferty Maths Dept. S3 Credit Pythagoras Theorem Twice
Learning Intention Success Criteria 1. Use the appropriate form of Pythagoras Theorem to solving harder problems. 2. Show all working. www.mathsrevision.com 1. To use knowledge already gained on Pythagoras Theorem to solve harder problems using Theorem twice. 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. Solving RealLife Problems
S3 Credit Problem : Find the length of h. www.mathsrevision.com
Find length BD first (BD ) = ( AD ) + ( AB )
2 2 2 B 15 A 13
19.85 h 12 D C (BD ) = 15 + 13
2 2 2 (BD )2 = 394
BD = 394
13 Feb 2012 BD = 19.85 Solving RealLife Problems
S3 Credit Problem : Find the length of length y. www.mathsrevision.com
Now find h (BC ) = (BD ) + (DC )
2 2 2 B 15 A 13
394 h 12 D C h 2 = (BD )2 + (DC )2
h 2 = (19.85)2 + (12)2 h 2 = 394 + 144 h = 538 h = 23.2 (to 1 d .p.)
13 Feb 2012 Solving RealLife Problems
S3 Credit Problem : Find the diagonal length of the cuboid AG. www.mathsrevision.com Find AH first ( AH ) = ( AD ) + (DH )
2 2 2 F B E A C G 7cm H 6cm ( AH )2 = 82 + 62
( AH )2 = 64 + 36 AH = 100 AH = 10cm
13 Feb 2012 10cm 8cm D Solving RealLife Problems
S3 Credit Problem : Find the diagonal length of the cuboid AG. www.mathsrevision.com Now find AG ( AG ) = ( AH ) + (HG )
2 2 2 F B E A C G 7cm H 6cm ( AG )2 = 102 + 72
( AG )2 = 100 + 49 ( AG )2 = 149 AG = 12.2cm ( to 1 d .p.)
13 Feb 2012 10cm 8cm D S3 Credit Pythagoras Theorem Now try Ex 5.1 Ch8 (page 154) www.mathsrevision.com
13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. S3 Starter Questions
S3 Credit 1.
www.mathsrevision.com True or f alse 1 1 1  = 6 7 42 2. 3. 4.
13 Feb 2012 Expand (1  w)(2 + 3w) Does ab2  a 2b f act orise t o b2a 2 (a  b) 2 5 1 1 5 3 Compiled by Mr. Lafferty Maths Dept. S3 Credit Converse of Pythagoras Theorem
Learning Intention Success Criteria 1. Apply the converse of Pythagoras Theorem to prove a triangle is rightangled. www.mathsrevision.com 1. To explain the converse of Pythagoras Theorem to prove a triangle is rightangled. 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. S3 Credit of Pythagoras Theorem Converse opposite, reverse
2 Converse Converse1 talk www.mathsrevision.com Converse Theorem states that if c a a +b = c
1. 2. 2 2 2 b Then triangle MUST be rightangled. Rightangle is directly opposite C. Hypotenuse 13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. S3 Credit Converse of Pythagoras Theorem
Problem : Is this triangle rightangled ? Explain Answer www.mathsrevision.com If it is then Pythagoras Theorem will be true
a 2 +b 2 = c 2 a +b = 9 + 6 = 8 1 +3 6 = 117
2 2 2 2 10cm 6 cm 9 cm c = 10 0
2 117 10 0 By the Converse Theorem, triangle is NOT rightangled
13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. S3 Credit Converse of Pythagoras Theorem
Problem : A picture frame manufacturer claims that his are rectangular is his claim true. www.mathsrevision.com If it is then Pythagoras Theorem will be true
a 2 +b 2 = c 2
2 2 a +b = 4 0 +3 0 = 16 0 0 + 9 0 0 = 2 5 0 0
2 2 40 cm 50cm c2 =2500
a 2 +b 2 = c 2
By the Converse Theorem, frame IS rectangular
13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. 30 cm S3 Credit Converse of Pythagoras Theorem Now try Ex 6.1 & 6.2 Ch8 (page 156) www.mathsrevision.com
13 Feb 2012 Compiled by Mr. Lafferty Maths Dept. ...
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