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Pythagoras Theorem

Pythagoras theorem is applicable in case of right-angled triangle. It says that, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
 
(Hypotenuse)2 = (Base)2 + (Perpendicular)2
a2 + b2 = c2
 
The smallest example is a = 3, b = 4 and c = 5. You can check that
32 + 42 = 9 + 16 = 25 = 52.

Sometimes we use the notation (a, b, c) to denote such a triple.

 

Notice that the greatest common divisor of the three numbers 3, 4 and 5 is 1. Pythagorean triples with this property are called primitive.

Proofs of Pythagoras Theorem

Proof 1
 
Description: P-322-3.tif

Now we start with four copies of the same triangle. Three of these have been rotated at 90°, 180°, and 270°, respectively. Each has the area ab/2. Let’s put them together without additional rotations so that they form a square with side c.
 
Description: P-322-4.tif
 
The square has a square hole with the side (a − b). By summing up its area (a − b)2 and 2ab, the area of the four triangles (4·ab/2), we get
C2 = (ab)2 + 2ab = a2 b2. QED

Proof 2
ABC is a right-angled triangle at B
To Prove: AC2 = AB2 + BC2
 
Description: P-322-5.tif

Construction: Draw BD ⊥ AC
 
Proof: ∆ADB ∼∆ABC (Property 8.5)
Description: 7255.png (Sides are proportional)
Or AB2 = AD × AC (1)
Also, ∆CDB ∼∆CBA
∴ Description: 7247.png
Description: 7240.png (Sides are proportional)
or BC2 = CD × CA
Adding (1) and (2)
AB2 + BC2 = AD × AC + CD × AC
= AC [AD + CD]
= AC × AC = AC2 QED.

Pythagorean Triplets

A Pythagorean triplet is a set of three positive whole numbers a b and c that are the lengths of the sides of a right triangle.
 
a2 + b2 = c2

It is noteworthy to see here that all of ab and c cannot be odd simultaneously. Either of a or b has to be even and c can be odd or even.

The various possibilities for ab and c are tabled below.
 
a b c
Odd Odd Even
Even Odd Odd
Odd Even Odd
Even Even Even

Some Pythagoras triplets are:
3 = 4 = 5 = (32+ 42 = 52)
 
5 = 12 = 13 = (52 + 122 = 132)
 
7 = 24 = 25 = (72 + 242 = 252)
 
8 = 15 = 17 = (82 + 152 = 172)
 
9 = 40 = 41 = (92 + 402 = 412)
 
11 = 60 = 61 = (112 + 602 = 612)
 
20 = 21 = 29 = (202 + 212 = 292)

 

Note : If each term of any pythagorean triplet is multiplied or divided by a constant (say P, P > 0) then the triplet so obtained will also be a pythagorean triplet. This is because if a2 + b2 = c2, then (Pa)2 + (Pb)2 = (Pc)2, where P > 0.

For example,
3 × 2    4 × 2    5 × 2    gives
6         8           10    (62 + 82 = 102)

Using Pythagoras theorem to determine the nature of triangle
 
If c2 = a2 + b2, then the triangle is right-angled triangle.
 
If c2 > a2 + b2, then the triangle is an obtuse-angled triangle.
 
If c2 < a2 + b2, then the triangle is an acute-angled triangle.

Mechanism to derive a Pythagorean triplet If the length of the smallest side is odd, assume the length of the smallest side = 5
 

Step 1 Take the square of 5 (length of the smallest side) = 25
 

Step 2 Break 25 into two parts P and Q, where P – Q = 1. In this case, P = 13 and Q = 12. Now these two parts P and Q along with the smallest side constitute pythagorean triplet.

However, there is another general formula for finding out all the primitive pythagorean triplets:
a = r2 − s2,
b = 2 rs,
c = r2 + s2,
r > s > 0 are whole numbers,
r − s is odd, and
The greatest common divisor of r and s is 1.

Table of small primitive Pythagorean triplets Here is a table of the first few primitive Pythagorean triplets:
 
r s a b c
2 1 3 4 5
3 2 5 12 13
4 1 15 8 17
4 3 7 24 25
5 2 21 20 29
5 4 9 40 41
6 1 35 12 37
6 5 11 60 61
7 2 45 28 53

Perimeter, area, inradius and shortest side

The perimeter P and area K of a Pythagorean triple triangle are given by
 
P = a + b + c = 2r(r + sd.
 
K = ab/2 = rs (r2 – s2d2.
 
Example-1
Two sides of a plot measure 32 m and 24 m and the angle between them is a right angle. The other two sides measure 25 m each and the other three angles are not right angles.
Description: P-324-1.tif
 
What is the area of the plot (in m2)?
  1. 768
  2. 534
  3. 696
  4. 684
Solution
The figure given above can be seen as
Description: P-324-2.tif
 
Since ABD is a right-angled triangle, so it will satisfy the Pythagoras theorem. And the triplet used here is – 3(×8), 4(×8) and 5(×8). Similarly the other part of the figure can also be bifurcated by drawing a perpendicular from C on BD.
 
So, the area of the plot is:
Area (∆ABD) + Area (∆CBD) Description: 7180.png × 24 × 32 + 2 × (1/2 × 20 × 15) = 684 m2
 
 
Example-2
A ladder of length 65 m is resting against a wall. If it slips 8 m down the wall, then its bottom will move away from the wall by N m. If it was initially 25 m away from it, what is the value of x?
Description: P-324-3.tif
  1. 60 m
  2. 39 m
  3. 14 m
  4. 52 m
Solution
Using Pythagorean triplets, (5, 12, 13),
⇒ h = 60
 
After it has slipped by 8 m, the new height = 52 m, and the length of the ladder = 65 m.
 
So 25 + x = 39 (3, 4, 5 triplet)
⇒ x = 14 m
 




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