Question

A square plot of land was divided into 16 equal parts by drawing lines parallel to its sides. how many rectangle (not equal) were formed in the resulting figure ? A) 64 B) 70 C) 40 D) 62

Answer 1

it is little difficult to show how much rectangle will be form. so, let's try to solve with help of shown diagram. here , a square AEUY is drawn and inside it, drawn 16 equal squares.

in first column, {ACHF}, {CEHJ}, {ADFI}, {BEGJ}, {AEFJ} = 5 rectangles

similarly , 2nd , 3rd and 4th columns , each have 5 rectangles.

so, total 5 × 4 = 20 rectangles.

in first row, {ABLK}, {KLVU}, {ABQP}, {FGVU}, {ABUV} = 5 rectangles

similarly, 2nd , 3rd and 4th rows , each have 5 rectangles

so, total 5 × 4 = 20 rectangles .

in 1st combination of two columns,
{ADNK}, {BEOL}, {AEOK} = 3 rectangles
similarly , 2nd combination of two columns = 3 rectangles
so, total 3 + 3 = 6 rectangles

in 1st combination of two rows ,
{ACRP}, {FHWU}, {ACWU} = 3 rectangles
similarly , 2nd combination of two rows , = 3
so, total 3 + 3 = 6 rectangles

now , see in FJTP , there are three rectangles
and similarly BDXV , there are three rectangles
so, total 3 + 3 = 6 rectangles .

see AETP , 2 rectangles
FJYU , 2 rectangles
so, total 2 + 2 = 4 rectangles

now total rectangles = 20 + 20 + 6 + 6 + 6 + 4
= 40 + 22 = 62

option (D) is correct.

Answer 2

Given :  A square plot of land was divided into 16 equal parts by drawing lines parallel to its sides.

To Find : how many rectangle (not equal sides) were formed in the resulting figure  

A) 64 B) 70 C) 40 D) 62

Solution:

Here we need to find rectangles - Squares

first finding squares :

Assume that given Square is 4 x 4 units

square is divided in 16 Equal parts by intersecting lines parallel to its sides .

Hence smallest square will be  1 x 1  sq unit

there will be 16 Squares of  1 x 1  units

there will be  9 square  of   2 x 2  units

there will be 4 square of  3 x 3  units

there will be  1 square  of 4 x 4  units  

Hence total number of square = 16 + 9 + 4 +  1  

= 30  

30 squares are formed in the resulting figure

Finding rectangles : Difficult to find rectangles one by one.

Easier method

there are 5 horizontal and 5 vertical lines.

we need to select 2 horizontal lines and 2 vertical lines to form a rectangle.

2 lines can be selected in ⁵C₂ = 10 ways

Hence total rectangles = 10 x 10 = 100   ( these includes squares also )

Hence  rectangle (not equal sides) were formed in the resulting figure

= 100 - 30

= 70

 formula  to find number of rectangles = (ⁿC₂)²  ( including squares)

( note : Direct/shortcut formula for finding square :  If divided into n x n  squares

then number of squares = n(n + 1) (2n + 1)/6

Here n = 4  =>  4(4+ 1) (2 * 4 + 1)/6  = 4 * 5 * 9 / 6 = 180/6 = 30 )

Learn More:

A square is divided 16 Equal parts by intersecting lines parallel to its side .how many squares are formed in the resulting figure?

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