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
: 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.
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 • 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
units and that the two legs are of length
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
Given: ABC is a right angle Triangle angle B =90 0 R.T.P:- AC 2 = AB 2 +BC 2 D Construction:- To draw BD AC . B C 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.
A 16 Hypotenuse B C 12
Solution : AC
(Pythagoras’ Theorem) AC
= 144 + 256 AC
= 400 AC = 20
Example 2. Find the length of diagonal d .
d 10 26 676 2 24 2
Application of Pythagoras’ Theorem
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.
16km N 12km ?
Solution : B 16 km In the figure, AB = 16 BC = 12 12 km C AC AC 2 2 = AB 2 + BC 2 = 16 2 + 12 2 AC 2 = 400 AC = 20 (Pythagoras’ Theorem) The displacement between the starting point and the destination point of the car is 20 km A
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.
Find the length of the shadow L.
Solution: 13 2 = 5 2 + L 2 L 2 = 13 2 L 2 = 144 - 5 2 (Pythagoras’ Theorem) L = 12 13 m 5 m L
. 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.