(i) Using the dots of a grid as the vertices, can you create a square that has an area of (a) 2 sq. units, (b) 3 sq. units, (c) 4 sq. units, and (d) 5 sq. units?
(ii) Suppose the grid extends indefinitely. What are the possible integer-valued areas of squares you can create in this manner?

Question

Bright Tutorials · Accepted Answer

(i) (a) 2 = 1
2
+ 1
2
Mark dots A, B, C, and D as shown.
Join AB, BC, CD, and DA.
Then ABCD is a square and area ABCD = 2 sq. units
(b) Square with area 3 units is not possible as 3 ≠ a
2
+ a
2
for any integer ‘a’.
(c) 4 = 2 × 2
Mark dots A, B, C, D as shown.
Join AB, BC, CD, DA.
Then ABCD is a square.
and ar ABCD = 2 × 2 = 4 sq. units
(d) (i)
Mark dots A, B, C, and D as shown.
Join A, B, C, and D
AB
2
= 2
2
+ 1
2
= 5
AB = √5 units
Hence, ABCD is a square with an area of 5 sq units.
(ii) Let the given value of area be x, where ‘x’ is an integer.
Then, x = a
2
+ b
2
, where ‘a’ and ‘b’ are integers or x is a perfect square, we can create squares with vertices as dots of the grid.

The Baudhayana-Pythagoras Theorem — Question 8