Introduction to Pythagorean Theorem.pptx
- 2. LESSON 3 Triangles • A triangle has three sides and three angles. • The three angles always add to 180°.
- 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
- 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: a2 + b2 = c2 Put in what we know: 52 + 122 = c2 Calculate squares: 25 + 144 = c2 169 = c2 Swap sides: c2 = 169 Square root of both sides: c = √169 Calculate: c = 13
- 16. LESSON 3 EXAMPLE 3 • Find the missing value: Start with: a2 + b2 = c2 Put in what we know: 92 + b2 = 152 Calculate squares: 81 + b2 = 225 Transpose 81 to the right side: b2 = 225 − 81 Calculate: b2 = 144 Square root of both sides: b = √144 Calculate: b = 12
- 17. LESSON 3 EXAMPLE 4 • Find the missing value: Start with: a2 + b2 = c2 Put in what we know: a2 + 62 = 102 Calculate squares: a2 + 36 = 100 Transpose 36 to the right side: a2 = 100 − 36 Calculate: a2 = 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: a2 + b2 = c2 Put in what we know: 72 + 92 = c2 Calculate squares: 49 + 81 = c2 Calculate: c2 = 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 a2 + b2 = c2 ? •a2 + b2 = 102 + 242 = 100 + 576 = 676 •c2 = 262 = 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 82 + 152 = 162 ? 82 + 152 = 64 + 225 = 289, but 162 = 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