Presentation on theme: "The Pythagoras Theorem"— Presentation transcript:
1The Pythagoras Theorem A tool for Right Angled Triangles
2Objectives:- General Objectives:-- To develop the mental ability of studentsTo develop the logical reasoning among the students
3Specific objective The students will be able to : learn the definition of a right triangle, hypotenuse, and the legs of a right triangleto find the length of an unknown side of a right triangle, if two sides are knownprove the Pythagoras Theoremrealize that there is nothing mysterious about the theorems from their textbooks, but that they actually have the ability to figure these things out for themselves
4Time Management P.K. Testing & Motivation 2 to 3 min. Presentation minGroup Formation minGroup activity minFeed back and assignment min.Home assignment min.
6Lesson Launch P.K. Testing The knowledge of students about various type of anglesTypes of trianglesProperties of similar triangles
7Statement of Pythagoras Theorem The famous theorem by Pythagoras deﬁned the relationship between the three sides of a right triangle.Pythagorean Theorem says that in a right triangle, the sum of the squares of the two right-angle sides will always be the same as the square of the hypotenuse
8You may note that in Δ ADB and Δ ABC Now, let us take a right triangle ABC, right angled at B. Let BD be the perpendicular to the hypotenuse AC .You may note that in Δ ADB and Δ ABC∠ A = ∠ A (Common)and ∠ ADB = ∠ ABC (Right)So, Δ ADB ~ Δ ABC (AA Similarity Criteria) (1)Similarly, Δ BDC ~ Δ ABC (AA Similarity Criteria) (2)BACD
9So, from (1) and (2), triangles on both sides of the perpendicular BD are similar to the whole triangle ABC.Also, since Δ ADB ~ Δ ABCand Δ BDC ~ Δ ABCSo, Δ ADB ~ Δ BDCThe above discussion leads to the following theorem :Theorem 1 : If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.
10Let us now apply this theorem in proving the Pythagoras Theorem: Pythagoras Theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
11Proof…… We are given a right triangle ABC right angled at B. We need to prove that AC2 = AB2 + BC2Let us draw BD ⊥ AC (see Fig).Now, Δ ADB ~ Δ ABC (Theorem 1)So, AD/AB = AB/AC (Sides are proportional)Or, AD.AC = AB (1)BACD
12Also, Δ BDC ~ Δ ABC (Theorem 1) So, CD/BC = BC/ACOr, CD.AC = BC (2)Adding (1) and (2),AD.AC + CD.AC = AB2 + BC2or, AC (AD + CD) = AB2 + BC2or, AC.AC = AB2 + BC2or, AC2 = AB2 + BC2Hence Proved.
13The above theorem was earlier given by an ancient Indian mathematician Baudhayan (about 800 B.C.) in the following form :The diagonal of a rectangle produces by itself the same area as produced by its both sides (i.e., length and breadth).For this reason, this theorem is sometimes also referred to as the Baudhayan Theorem.
14A car drives 20 miles due east and then 45 miles due south. To the nearest hundredth of a mile,how far is the car from its starting point?20 milesx45 miles
15Group ActivityG1. A ladder is placed against a wall s.t. its foot is at a distance of 24 m from the wall and its top reaches a window 7m above the ground. Find the length of the ladder.7m?24m
16G2. Determine that which are the sides of a right angled triangle 1 G2.Determine that which are the sides of a right angled triangle 1.3cm,4cm,5cm. 2.13m,12m,5m. 3.3cm,8cm,6cm.
17G3.Tick the correct answer and justify: in triangle ABC ,AB=12m,AC=13m and BC=5m.The angle B is a b.60 c d.80
18Conclusion :- Hene we have find from all activities that In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
19Home Assignment:-Q.No. 1 to 8 , From exercise 6.4 will be home work to all of you. Firstly try these question at home yourself , If you are not able to solve them then we will take these problems in class.
20This Lesson Plan has been compiled and presented by : Subodh Kumar T GT (NM) GHS P A NOHAshok Sharma TGT (NM) GMS LALHRISanjeev Thakur T GT (NM) GMS JALGRANSanjiv Lath TGT (NM) GMS BHANJALRavi Rana TGT (NM) GMS PALKWAHKuldeep Chandel TGT (NM) GSSS POLIAN PROTANDISTT. UNA