Question

how many rectangles are in this figure​

Answer 1

Answer: 45 rectangles

Here the squares are also considered as rectangles.

In this 2 × 5 grid,

Consider 1 × 1.

$\begin{tabular}{|c|}\cline{1-2}&\cline{1-2}\end{tabular}$

⇒  When counted, 2 pieces each respecting to the horizontal side are along the row and 5 pieces each respecting to the vertical side are along the column.

⇒  Thus there are 2 × 5 = 10   1 × 1  rectangles.

Consider 1 × 2.

$\begin{tabular}{|c|}\cline{1-2}&\cline{1-2}\\ \cline{1-2}\end{tabular}$

⇒  When counted, 2 pieces each respecting to the horizontal side are along the row and 4 pieces each respecting to the vertical side are along the column.

⇒  Thus there are 2 × 4 = 8   1 × 2  rectangles.

Consider 1 × 3.

$\begin{tabular}{|c|}\cline{1-2}&\cline{1-2}\\ \cline{1-2}&\cline{1-2}\end{tabular}$

⇒  When counted, 2 pieces each respecting to the horizontal side are along the row and 3 pieces each respecting to the vertical side are along the column.

⇒  Thus there are 2 × 3 = 6   1 × 3  rectangles.

Consider 1 × 4.

$\begin{tabular}{|c|}\cline{1-2}&\cline{1-2}\\ \cline{1-2}&\cline{1-2}&\cline{1-2}\end{tabular}$

⇒  When counted, 2 pieces each respecting to the horizontal side are along the row and 2 pieces each respecting to the vertical side are along the column.

⇒  Thus there are 2 × 2 = 4   1 × 4  rectangles.

Consider 1 × 5.

$\begin{tabular}{|c|}\cline{1-2}&\cline{1-2}\\ \cline{1-2}&\cline{1-2}&\cline{1-2}&\cline{1-2}\end{tabular}$

⇒  When counted, 2 pieces each respecting to the horizontal side are along the row and 1 pieces each respecting to the vertical side are along the column.

⇒  Thus there are 2 × 1 = 2   1 × 5  rectangles.

Consider 2 × 1.

$\begin{tabular}{|c|c|}\cline{1-2}&&\cline{1-2}\end{tabular}$

⇒  When counted, 1 piece respecting to the horizontal side are along the row and 5 pieces each respecting to the vertical side are along the column.

⇒  Thus there are 1 × 5 = 5   2 × 1  rectangles.

Consider 2 × 2.

$\begin{tabular}{|c|c|}\cline{1-2}&&\cline{1-2}&&\cline{1-2}\end{tabular}$

⇒  When counted, 1 piece respecting to the horizontal side are along the row and 4 pieces each respecting to the vertical side are along the column.

⇒  Thus there are 1 × 4 = 4   2 × 2  rectangles.

Consider 2 × 3.

$\begin{tabular}{|c|c|}\cline{1-2}&&\cline{1-2}&&\cline{1-2}&&\cline{1-2}\end{tabular}$

⇒  When counted, 1 piece respecting to the horizontal side are along the row and 3 pieces each respecting to the vertical side are along the column.

⇒  Thus there are 1 × 3 = 3   2 × 3  rectangles.

Consider 2 × 4.

$\begin{tabular}{|c|c|}\cline{1-2}&&\cline{1-2}&&\cline{1-2}&&\cline{1-2}&&\cline{1-2}\end{tabular}$

⇒  When counted, 1 piece respecting to the horizontal side are along the row and 2 pieces each respecting to the vertical side are along the column.

⇒  Thus there are 1 × 2 = 2   2 × 4  rectangles.

Consider 2 × 5.

$\begin{tabular}{|c|c|}\cline{1-2}&&\cline{1-2}&&\cline{1-2}&&\cline{1-2}&&\cline{1-2}&&\cline{1-2}\end{tabular}$

⇒  The given grid is of 2 × 5. Thus there is only 1 × 1 = 1.

Total no. of rectangles

→  10 + 8 + 6 + 4 + 2 + 5 + 4 + 3 + 2 + 1

→  45

Thus there are a total of 45 rectangles.

A shortcut...

We found that,

  ⇒  No. of 1 × 1 grid in 2 × 5 rectangle = 2 × 5

  ⇒  No. of 1 × 2 grid in 2 × 5 rectangle = 2 × 4

  ⇒  No. of 1 × 3 grid in 2 × 5 rectangle = 2 × 3

  ⇒  No. of 1 × 4 grid in 2 × 5 rectangle = 2 × 2

  ⇒  No. of 1 × 5 grid in 2 × 5 rectangle = 2 × 1

  ⇒  No. of 2 × 1 grid in 2 × 5 rectangle = 1 × 5

  ⇒  No. of 2 × 2 grid in 2 × 5 rectangle = 1 × 4

  ⇒  No. of 2 × 3 grid in 2 × 5 rectangle = 1 × 3

  ⇒  No. of 2 × 4 grid in 2 × 5 rectangle = 1 × 2

  ⇒  No. of 2 × 5 grid in 2 × 5 rectangle = 1 × 1

Thus, total no. of rectangles...

$(2\times 5)+(2\times 4)+(2\times 3)+(2\times 2)+(2\times 1)+(1\times 5)+(1\times 4)+(1\times 3)+(1\times 2)+(1\times 1) \\ \\ \\ 2(5+4+3+2+1)+1(5+4+3+2+1) \\ \\ \\ (2+1)(5+4+3+2+1) \\ \\ \\ 3 \times 15=\ \large \textbf{45}$

From this method, we get 'four' shortcuts...!!!

$\large \textit{The no. of rectangles in an ` m \times n ' \ grid is,} \\ \\ \\ 1.\ (1+2+3+...+m)(1+2+3+...+n) \\ \\ \\ 2.\ \displaystyle \left(\frac{m(m+1)}{2}\right)\left(\frac{n(n+1)}{2}\right) \\ \\ \\ 3.\ ^{m+1}\!C_2 \ \cdot \ ^{n+1}\!C_2 \\ \\ \\ 4.\ ^{m+1}\!C_{m-1} \ \cdot \ ^{n+1}\!C_{n-1}$

This product can also be written as,

$\displaystyle \sum_{k=1}^{m}k \ \cdot \ \sum_{k=1}^{n}k$

Answer 2

Answer:

There are 45 rectangles in the figure.. ..