To day we are teaching about PYTHAGOREAN THEOREM
Pythagorean Theorem Pythagoras (~580-500 B.C.) He was a Greek philosopher responsible for important developments in mathematics, astronomy and the theory of music.
1. cut a triangle with base 4 cm and height 3 cm 4 cm 3 cm 2. measure the length of the hypotenuse Now take out a square paper and a ruler. 5 cm 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
Consider a square PQRS with sides a + b c c c c Now, the square is cut into - 4 congruent right-angled triangles and - 1 smaller square with sides c Proof of Pythagoras’ Theorem P Q R S a a a a b b b b
Area of square ABCD = ( a + b ) 2 Area of square PQRS a 2 + 2ab + b 2 = 2ab + c 2 a 2 + b 2 = c 2 a + b a + b A B C D b b a b b a a a c c c c P Q R S = 4 + c 2
Theorem states that: "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." The Pythagorean Theorem asserts that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: a 2 + b 2 = c 2 The figure above at the right is a visual display of the theorem's conclusion. The figure at the left contains a proof of the theorem, because the area of the big, outer, green square is equal to the sum of the areas of the four red triangles and the little, inner white square: c 2 = 4(ab/2) + (a - b) 2 = 2ab + (a 2 - 2ab + b 2 ) = a 2 + b 2
Animated Proof of the Pythagorean Theorem Below is an animated proof of the Pythagorean Theorem. Starting with a right triangle and squares on each side, the middle size square is cut into congruent quadrilaterals (the cuts through the center and parallel to the sides of the biggest square). Then the quadrilaterals are hinged and rotated and shifted to the big square. Finally the smallest square is translated to cover the remaining middle part of the biggest square. A perfect fit! Thus the sum of the squares on the smaller two sides equals the square on the biggest side. Afterward, the small square is translated back and the four quadrilaterals are directly translated back to their original position. The process is repeated forever.
Pythagorean Theorem Over 2,500 years ago, a Greek mathematician named Pythagoras developed a proof that the relationship between the hypotenuse and the legs is true for all right triangles . In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs." This relationship can be stated as: and is known as the Pythagorean Theorem a, b are legs. c is the hypotenuse (across from the right angle). There are certain sets of numbers that have a very special property. Not only do these numbers satisfy the Pythagorean Theorem, but any multiples of these numbers also satisfy the Pythagorean Theorem For example : the numbers 3, 4, and 5 satisfy the Pythagorean Theorem. If you multiply all three numbers by 2 (6, 8, and 10), these new numbers ALSO satisfy the Pythagorean theorem.
If we think about a right triangle we know of course that one of the angles is a right angle. We also know that the other two angles are acute angles (why?). In fact we know that the other two angles are complementary angles. Therefore there is a relationship between the sizes of the angles that the two acute angles have measures that add up to ninety degrees. What about sides? Is there a relationship between the sides of a right triangle? We know from previous lessons that if we have the lengths of just two of the sides we can construct the triangle so it is enough to know the lengths of two sides to determine the length of the third side. We shall now try to figure out the relationship. We shall, to make it easy to communicate assume that the length of the hypotenuse is c units and that the two legs are of length a and b units. So according to the Pythagorean Theorem, the area of square A, plus the area of square B should equal the area of square C. The special sets of numbers that possess this property are called Pythagorean Triples. The most common Pythagorean Triples are: 3, 4, 5 5, 12, 13 8, 15, 17
The Pythagorean Theorem
The Pythagorean Theorem is one of Euclidean Geometry's most beautiful theorems. It is simple, yet obscure, and is used continuously in mathematics and physics. In short, it is really cool.
This first method is one of the ways the Pythagoreans would have proved the theorem. Unfortunately, it lacks glamour. In the following picture let ABC be a right triangle and BD be a segment drawn perpendicular to AC.
Since the triangles are similar, the sides must be of proportional lengths.
AB/AD=AC/AB, or AB x AB = AD x AC
BC/CD=AC/BC, or BC x BC= AC x CD
Then, adding the two together, BC^2 + AB^2 = (AD + DC) x AC= AC^2
Pythagorean Theorem in text book of 10 th class Given:- ABC is a right angle Triangle . angle B =90 0 R.T.P:- AC 2 = AB 2 +BC 2 Construction:- To draw BD AC . A B C D Proof:- In Triangles ADB and ABC AngleA=Angle A (common) Angle ADB=Angle ABC (each 90 0 ) ADB ~ ABC ( A.A corollary ) So that AD/AB=AB/AC (In similar triangles corresponding sides are proportional ) AB 2 = AD X AC _________(1) Similarly BC 2 = DCXAC _________(2) Adding (1)&(2) we get AB 2 +BC 2 = AD X AC + DCXAC = AC (AD +DC) = AC . AC =AC 2 There fore AB 2 +BC 2 =AC 2
Example 1. Find the length of AC. AC 2 = 12 2 + 16 2 (Pythagoras’ Theorem) AC 2 = 144 + 256 AC 2 = 400 AC = 20 A C B 16 12 Solution : Hypotenuse
Example 2. Find the length of diagonal d . Solution: 10 24 d d 2 = 10 2 + 24 2 (Pythagoras’ Theorem)
16km 12km 1.A car travels 16 km from east to west. Then it turns left and travels a further 12 km. Find the displacement between the starting point and the destination point of the car. N ? Application of Pythagoras’ Theorem
Solution : In the figure, AB = 16 BC = 12 AC 2 = AB 2 + BC 2 (Pythagoras’ Theorem) AC 2 = 16 2 + 12 2 AC 2 = 400 AC = 20 The displacement between the starting point and the destination point of the car is 20 km 16 km 12 km A B C
2. The height of a tree is 5 m. The distance between the top of it and the tip of its shadow is 13 m. Solution: 13 2 = 5 2 + L 2 (Pythagoras’ Theorem) L 2 = 13 2 - 5 2 L 2 = 144 L = 12 Find the length of the shadow L. L 5 m 13 m
Exercise . 1.Find the length of the hypotenuse for right triangles with legs, and sketch the triangles a).3 and 4? b).5 and 12? c).5.2 and 10.5? 2.Find the lengths of the other leg of right triangles if one leg =6 and the hypotenuse =2,8.3 and 7 in each case. 3. A right triangle with legs equal to 5cm and 12cm. What is the length of the hypotenuse? . 4. In your own words, explain what the Pythagorean Theorem states.
Text book of class X Subject:Mathematics Search engine:www.google.com Prepared by S.Jaya Prada