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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.
[Table_1]
(b) Three horizontal lines, $AB$, $CD$ and $EF$ are drawn inside the rectangle $PQRS$.
Complete the table to show all ten rectangles.
[Table_2]
(c) Four horizontal lines are drawn inside the rectangle.
Find the total number of rectangles.
(d) Complete the table.
[Table_3]
(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.
537
511
505
One vertical line, $AB$, is drawn inside rectangle $PQRS$.
The total number of rectangles is 3.
They are $PABS$, $PQRS$ and $AQRB$.
[Image_1: Rectangle with vertical line]
(a) Two vertical lines are drawn inside a rectangle.
Find the total number of rectangles. ..........................................................
(b) Complete the table.
[Table_1:
\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline Number \text{ of vertical} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \text{lines inside a rectangle} \\ \hline \text{Total number of} & 3 & & & & & & & \\ \text{rectangles} \\ \hline \end{array}]
(c) What is the connection between the table in \text{question 1(d)} and the table in \text{question 2(b)}?
562
546
527
12 vertical lines are drawn inside a rectangle.
Show that the total number of rectangles is given by the calculation $\frac{{12^2 + 3 \times 12 + 2}}{2}$.
582
518
539
(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.
509
535
535
When 30 horizontal lines are drawn inside a rectangle, find the total number of rectangles.
519
592
533
