Introduction to Pythagorean Theorem.pptx

Introduction to Pythagorean Theorem.pptx

1. TRIANGLES LESSON 3

2. LESSON 3 Triangles A triangle has three sides and three angles. The three angles always add to 180°.

3. Types of Triangles

4. LESSON 3 RIGHT TRIANGLE A right-angled triangle (also called a right triangle) is a triangle with a right angle (90°) in it. The  little square  in the corner tells us it is a right angled triangle

5. LESSON 3 There are two types of right angled triangle: Isosceles right-angled triangle One right angle Two other  equal  angles always of  45° Two equal sides Scalene right-angled triangle One right angle Two other  unequal  angles No equal sides

6. LESSON 3 Area The area is  half of the base times height . "b" is the distance along the base "h" is the height Area = ½ × b × h

7. Introduction to Pythagorean Theorem LESSON 3

8. LESSON 3 TARGET/S proves the Pythagorean Theorem. solves problems that involve right triangles.

9. LESSON 3 Introduction to Pythagorean Theorem Over 2000 years ago there was an amazing discovery about triangles: When a triangle has a right angle (90°) and squares are made on each of the three sides,

10. Introduction to Pythagorean Theorem Over 2000 years ago there was an amazing discovery about triangles: When a triangle has a right angle (90°) and squares are made on each of the three sides, then the biggest square has the  exact same area  as the other two squares put together.

11. LESSON 3 Introduction to Pythagorean Theorem It is called "Pythagoras' Theorem " and can be written in one short equation: Note: c  is the  longest side  of the triangle c - hypotenuse a  and  b  are the other two sides

12. LESSON 3 Introduction to Pythagorean Theorem The longest side of the triangle is called the "hypotenuse"

13. LESSON 3 Pythagorean Theorem In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

14. LESSON 3 EXAMPLE 1 A "3,4,5" triangle has a right angle in it. Let's check if the areas are the same: 3² + 4² = 5² Calculating this becomes: 9 + 16 = 25

15. LESSON 3 EXAMPLE 2 Find the missing value: Start with: a 2  + b 2  = c 2 Put in what we know: 5 2  + 12 2  = c 2 Calculate squares: 25 + 144 = c 2 169 = c 2 Swap sides: c 2  = 169 Square root of both sides: c = √169 Calculate: c = 13

16. LESSON 3 EXAMPLE 3 Find the missing value: Start with: a 2  + b 2  = c 2 Put in what we know: 9 2  + b 2  = 15 2 Calculate squares: 81 + b 2  = 225 Transpose 81 to the right side: b 2  = 225 − 81 Calculate: b 2  = 144 Square root of both sides: b = √144 Calculate: b = 12

17. LESSON 3 EXAMPLE 4 Find the missing value: Start with: a 2  + b 2  = c 2 Put in what we know: a 2  + 6 2  = 10 2 Calculate squares: a 2 + 36 = 100 Transpose 36 to the right side: a 2  = 100 − 36 Calculate: a 2  = 64 Square root of both sides: a = √64 Calculate: a = 8 a 6 10

18. LESSON 3 EXAMPLE 5 Find the missing value: Start with: a 2  + b 2  = c 2 Put in what we know: 7 2  + 9 2  = c 2 Calculate squares: 49 + 81 = c 2 Calculate: c 2  = 130 Square root of both sides: c = √130 Calculate: c = 11.40 9 7 ?

19. LESSON 3 Why Is This Useful? If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. But remember it only works on right angled triangles.

20. LESSON 3 EXAMPLE 3 Does this triangle have a Right Angle? Does a 2  + b 2  = c 2  ? a 2  + b 2  = 10 2  + 24 2  = 100 + 576 =  676 c 2  = 26 2  =  676 They are equal, so ... Yes, it does have a Right Angle.

21. LESSON 3 EXAMPLE 4 Does an 8, 15, 16 triangle have a Right Angle? Does 8 2  +  15 2  =  16 2  ? 8 2  + 15 2  = 64 + 225 =  289 , but 16 2  =  256 So, NO, it does not have a Right Angle.

22. References E-Math 9 - Work Text in Mathematics (Rex Book Store) Math Ideas and Life Applications 9 - Second Edition ( Abiva ) Spiral Math 9 – ( Trinitas Publishing Inc.) https://www.mathsisfun.com/triangle.html https://www.mathsisfun.com/right_angle_triangle.html https://www.mathsisfun.com/pythagoras.html