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This investigation looks for a method to find the number of rectangles when you draw horizontal and vertical lines inside a rectangle.
One horizontal line, $AB$, is drawn inside a rectangle $PQRS$.
The total number of rectangles is 3.
They are $PQBA$, $PQRS$ and $ABRS$.
(a) Another line $CD$ is drawn inside the rectangle $PQRS$.
The total number of rectangles is now 6.
Four of the 6 rectangles are $PQBA$, $PQDC$, $PQRS$ and $ABDC$.
Complete the table to show the other two rectangles.
| PQBA | PQDC | PQRS |
|------|------|------|
| ABDC | | |
(b) Three horizontal lines, $AB$, $CD$ and $EF$ are drawn inside the rectangle $PQRS$.
Complete the table to show all ten rectangles.
| PQBA | | PQRS |
|------|------|------|
| ABDC | ABRS | |
| CDRS | | |
(c) Four horizontal lines are drawn inside the rectangle.
Find the total number of rectangles.
(d) Complete the table.
| Number of horizontal lines inside the rectangle | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----------------------------------------------|---|---|---|---|---|---|---|---|
| Total number of rectangles | 3 | 6 | 10| | | | |36 |
(e) The numbers in the bottom row of the table in part (d) form a sequence.
Write down the mathematical name of these numbers.
(f) Ten horizontal lines are drawn inside the rectangle.
Find the total number of rectangles.
584
518
555
One vertical line, $AB$, is drawn inside rectangle $PQRS$.
The total number of rectangles is 3.
They are $PABS$, $PQRS$, and $AQRB$.
(a) Two vertical lines are drawn inside a rectangle.
Find the total number of rectangles.
..............................................................
(b) Complete the table.
[Table_1]
| Number of vertical lines inside a rectangle | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| Total number of rectangles | 3 |
(c) What is the connection between the table in question 1(d) and the table in question 2(b)?
......................................................................................................................................................
500
516
551
12 vertical lines are drawn inside a rectangle.
Show that the total number of rectangles is given by the calculation $\frac{12^2 + 3 \times 12 + 2}{2}$.
568
594
547
(a) When $n$ vertical lines are drawn inside a rectangle the total number of rectangles, $T$, is $$ T = \frac{1}{2}n^2 + an + b, $$ where $a$ and $b$ are constants.
Find the value of $a$ and the value of $b$. Use your answers to write down the formula for $T$.
$a = \text{.........................................}$
$b = \text{.........................................}$
$T = \text{..........................................................}$
(b) Use your formula in part (a) to show that when 7 vertical lines are drawn inside a rectangle, the number of rectangles is 36.
(c) Calculate how many vertical lines are drawn when there are 231 rectangles.
..........................................................
535
561
506
When 30 horizontal lines are drawn inside a rectangle, find the total number of rectangles.
501
573
549
