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activity
Take a card sheet and draw a right-angled triangle ABC, right angled at C on the sheet.
Consider the lengths of the side BC, CA, and AB be a, b, and c respectively.
Draw squares on the sides BC, CA, and AB as shown in the given figure (i). A
Now, cut the DABC and the all three squares separately from the sheet and cut 7 more b c
identical triangles of the DABC from the sheet. Now, we have eight identical copies of C a B
DABC altogether.
Paste four identical triangles around the sides of the square of side AB (c) on another
sheet shown in figure (ii). Therefore, we get a square of side (a + b). Fig (i)
Now, paste the squares of sides of BC and CA of measure a and b respectively and 4 b a
remaining identical copies of the DABC on the paper sheet as shown in figure (iii). Thus, b
we again get a square of side (a + b). a c
From figure (ii) and (iii), it is clear that c
Area of square in figure (ii) = Area of square in figure (iii) = (a + b) 2 c c
Now, remove all the triangles from both the figures (ii) and (iii). b a
Then only the square of side c will be left in figure (ii) and two squares of sides a and
b will be left in figure (iii). a Fig (ii) b
As from the above the area of figure (ii) = area of figure (iii). b a
c
And, we have removed same areas from both the figures, Then the remaining area of both b b b
the figures will be same.
2
2
2
Therefore, we get c = a + b .
Thus, we can say that the square of the hypotenuse (c) is equal to the sum of the squares a a c a
of other two sides a and b. Hence, the Pythagoras Property is verified.
a Fig (iii) b
Converse of the Pythagoras Property

Let us take a right triangle ABC, right angle at C of sides BC = 3 cm, AC =
4 cm, and AB = 5 cm and apply the Pythagoras property on it. Clearly, the A
measure of longest side is 5 cm. Therefore, c 25
2
2
c = 5 ⇒ c = 5 = 5 × 5 = 25 16 b 5
2
2
a = 3 ⇒ a = 3 = 3 × 3 = 9 4 C a B
2
2
b = 4 ⇒ b = 4 = 4 × 4 = 16 9
2
2
2
2

a + b = 9 + 16 = 25 Or c = a + b 2 3
So, we can say, it is a right-angled triangle of measure 3 cm, 4 cm and 5 cm.
Hence, if the Pythagoras property holds, then the triangle must be a right-angled. It represents
the converse of the Pythagoras property.
A Pinch of History
Baudhayana's theorem, known centuries before Pythagoras, is an ancient
Indian mathematical principle outlined in the Baudhayana Sulba Sutra. It Diagonal
states that the areas produced separately by the length and the breadth of a
rectangle together equal the area produced by the diagonal. This concept, much Side 1
like the Pythagoras theorem, was used to solve practical geometric problems,
especially in altar construction, where precise measurements were crucial.
Side 2
The theorem highlights the advanced mathematical knowledge present in
India around 800 BC, well before its recognition in Greece.

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