The Pythagoras Theorem A tool for Right Angled Triangles

Objectives:-

• • • General Objectives:- To develop the mental ability of students To develop the logical reasoning among the students

Specific objective

• • • • • The students will be able to : learn the definition of a right triangle, hypotenuse, and the legs of a right triangle to find the length of an unknown side of a right triangle, if two sides are known prove the Pythagoras Theorem realize 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

Time Management

1. P.K. Testing & Motivation 2 to 3 min.

2. Presentation 7min 3. Group Formation 2 min 4. Group activity 11min 5. Feed back and assignment 10min.

6. Home assignment 2min.

Material Required

1. Chart 2. Scale 3. Scissor/Blade 4. Compass 5. Divider

Lesson Launch 1. P.K. Testing – The knowledge of students about various type of angles – Types of triangles – Properties of similar triangles

• • 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

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) B A D C

So, from (1) and (2), triangles on both sides of the perpendicular BD are similar to the whole triangle ABC.

Also, since Δ ADB ~ Δ ABC and Δ BDC ~ Δ ABC So, Δ ADB ~ Δ BDC The 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.

Let 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.

Proof……

We are given a right triangle ABC right angled at B. We need to prove that AC 2 = AB 2 + BC 2 Let 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 2 B (1) C A D

Also, Δ BDC ~ Δ ABC (Theorem 1) So, CD/BC = BC/AC Or, CD.AC = BC 2 (2) Adding (1) and (2), AD.AC + CD.AC = AB 2 + BC 2 or, AC (AD + CD) = AB 2 + BC 2 or, AC.AC = AB 2 + BC 2 or, AC 2 = AB 2 + BC 2 Hence Proved.

The 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.

A 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 miles x 45 miles a 2  b 2 400 2 20   45 2 2025 2425     c 2 x 2 x 2 x 2 2425  x 2 x  2425 x  49.24

Group Activity

• G1. 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

sides of a right angled triangle 1.3cm,4cm,5cm.

2.13m,12m,5m.

3.3cm,8cm,6cm.

G3.Tick the correct answer and justify: in triangle ABC ,AB=12m,AC=13m and BC=5m.The angle B is a.120 b.60

c.90 d.80

Conclusion :-

• • 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.

Home 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.

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