Pythagorean Theorem a2 + b2 = c2 Reem Alabdulkarim

Objectives At the end of the topic, the student must be able to understand and demonstrate the following: How to recognize a right-angled triangle How to solve the equation associated with the Pythagorean Theorem How to identify and label the three sides of the right triangle according to the Pythagorean Theorem. The applications of Pythagorean Theorem.

History of Pythagoras Pythagoras was a Greek mathematician who lived between 570 to 495 BC and is credited with the Pythagorean theorem’s framework. his studies are classified under a school of knowledge known as Pythagoreanism.

The three figures do obey the Pythagorean theorem. Right triangles A These triangles are right angled triangles judging by the angle subtended by the two sides which is 90⁰ . The three figures do obey the Pythagorean theorem. 90 90 90 C

They do not obey the Pythagorean theorem. Non-right triangle These are not right angled triangles and the angles subtended by any of their two sides is not 90⁰. They do not obey the Pythagorean theorem. Obtuse Angle Triangle Acute Angle Triangle

hypotenuse The hypotenuse is a side opposite to the right angle of a triangle and joins the ends of the two sides that subtend the angle. Hypotenuse C Leg of right triangle B Leg of right triangle A Right angle

Pythagorean theorem The theorem is a relation between the three sides of a right-angled triangle in Euclidean geometry. It states that the square of the hypotenuse (c) is equal to the sum of the squares of sides adjacent to the right angle (a) and (b). The equation then gives a2 + b2 = c2 c b a

3, 4, 5 ratio. Considering a triangle X of ratio 3:4:5, the area of the adjacent squares are 9:16:25 respectively. Mathematically 25= 16+9, therefore the area adjacent to the hypotenuse equals to the sum of the areas adjacent to the two sides. If the sides 3,4 and 5 are represented by a, b and c and there areas by , and then the equation conforms to a2 + b2 = c2 32 + 42 = 52 9 + 16 = 25 5 3 4

Real world examples This section seeks to show how the theorem is applied in almost every aspect of our lives. painting on a wall: when painting on a wall, painters incline their ladders at particular angles and this relates to the height of the wall. For instance, if the wall to be painted is 8 meters high, the painter will require a ladder that is 10meters. This is deduced from placing the ladder 6 meters from the wall which gives a2 + b2 = c2 d2 + h2 = c2 62 + 82 =? 36+64=√100 =10 102 82 62

Real world examples while shopping for items say for example, a computer and your desk is 22 inches in length, it is possible to calculate the dimensions of the computer screen. If you walk inside a store, you will find a monitor with dimensions of 16x10 inches. Applying the theorem 162 + 102 = 182 18 inches is well within the bracket of a 22 inches desk cabin.

Which of the three triangles are right angled?. Right angle self test 1 Which of the three triangles are right angled?. 3 2

Correct Right angle The triangle number 1 is a right angled triangle. 3 2

Practice Exercise: Case 1 Case 2 Case 3 Find the missing result of A, B , and C in the triangles B=? B=5 B=12 C=? C=7 C=13 A=? A=9 A=3 Case 1 Case 2 Case 3

Case 1 Case 1 Case 1 C=5 C=13 C=? B=? B=5 B=12 A=? A=4 A=9 a2 + b2 = c2 42 + b2 = 52 16+b2 =25 b2=9 √9=√b2 B=3 a2 + b2 = c2 a2 + 52 = 132 a2+25=169 A2=144 √144=√a2 a=12 a2 + b2 = c2 92 + 122 = c2 81+144=c2 225=c2 √225=√c2 C=15

Review From the above we can deduce that the theorem is both viable and applicable. The various tests and examples aid in building onto the knowledge of the theorem.