Find the value of 15 by geometric method.

We have 15 = 3 × 5 . Steps of Construction: (a) Draw a line segment B P and from it cut off a length B D equal to 3 cm and a length D C equal to 5 cm . (b) Taking B C as diameter draw a semicircle. (c) Draw a perpendicular at D on B C which intersects the circumference of the semicircle at A . Then, the line segment A D is the mean proportion of B D and D C . So, A D 2 = B D × D C ⇒ A D = B D × D C ⇒ A D = 3 × 5 cm = 15 cm By scale, we measure and obtain A D = 3 . 9 cm (approx.). Therefore, 15 = 3 . 9 (approx.).

Find the value of 35 by geometric method.

We have 35 = 5 × 7 . Steps of Construction: (a) Draw a line segment B P and from it cut off a length B D equal to 5 cm and a length D C equal to 7 cm . (b) Taking B C as diameter draw a semicircle. (c) Draw a perpendicular at D on B C which intersects the circumference of the semicircle at A . Then, the line segment A D is the mean proportion of B D and D C . So, A D 2 = B D × D C ⇒ A D = B D × D C ⇒ A D = 5 × 7 cm = 35 cm By scale, we measure and obtain A D = 5 . 9 cm (approx.). Therefore, 35 = 5 . 9 (approx.).

Construct a square each of whose side is 20 cm .

We are given to construct a square each of whose side is 20 cm . We know that 20 = 5 × 4 . So, first, we shall construct a rectangle whose length is 5 cm and breadth is 4 cm . So, the area of this rectangle will be 5 × 4 = 20 cm 2 . Then, we shall construct a square whose area will be the same ( 20 cm 2 ), so that the length of its side will be 20 cm . Steps of Construction : a) Draw a rectangle A B C D whose length is 5 cm and breadth is 4 cm . b) Produce A B and from the produced portion cut off length B E equal to B C . c) Taking A E as diameter, draw a semicircle. C B is produced to meet the semicircle at F . d) Produce B E and from the produced portion cut off a length B H equal to B F . e) Taking F and H as centre and B F as radius, two circular arcs are drawn which meet at G . f) Join F and G , H and G . Then, B F G H is the required square whose side is 20 cm .