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(a) Draw the next trapezium in the sequence. [1]
(b) Complete the table.
[Table_1]
$$ \begin{array}{ccc} \text{Shorter parallel side length (x)} & \text{Longer parallel side length} & \text{Perimeter} \\ 1 & 2 & 5 \\ 2 & & \\ 3 & 4 & 9 \\ 4 & & \end{array} $$ [2]
(c) Write down an expression, in terms of \( x \), for the length of the longer parallel side.
.................................................. [1]
(d) Find an expression, in terms of \( x \), for the perimeter. Give your answer in its simplest form.
.................................................. [2]
520
570
513
The diagram shows another sequence of trapeziums.
Each trapezium has sloping side length 2.
Both parallel sides increase in length by 1 each time.
!
(a) Draw the trapezium with sloping side length 2 and $x = 5$. [1]
(b) Complete the table.
| Shorter parallel side length $(x)$ | Longer parallel side length | Perimeter |
|---|---|---|
| 1 | 3 | 8 |
| 2 | 4 | |
| 3 | 5 | |
| 4 | 6 | 14 |
| 5 |
(c) Write down an expression, in terms of $x$, for the length of the longer parallel side.
........................................................ [1]
(d) Find an expression, in terms of $x$, for the perimeter.
Give your answer in its simplest form.
........................................................ [2]
569
591
575
In another sequence each trapezium has sloping side length 3.
The length of the shorter parallel side is $x$.
(a) Find an expression, in terms of $x$, for the length of the longer parallel side of the trapeziums with sloping side length 3.
You may use the grid below to help you. [2]
(b) Show that an expression, in terms of $x$, for the perimeter is $2x + 9$. [1]
565
540
585
In each trapezium:
- the shorter parallel side length is $x$
- each sloping side length is $y$.
(a) Write down an expression, in terms of $x$ and $y$, for the length of the longer parallel side.
You may use your expressions from Questions 1(c), 2(c) and 3(a) to help you. ....................................................... [1]
(b) Find an expression, in terms of $x$ and $y$, for the perimeter. Give your answer in its simplest form. ....................................................... [2]
(c) Show that your expression in part (b) gives the correct result when $x = 2$ and $y = 5$. ....................................................... [3]
(d) A trapezium has shorter parallel side length $x$ and sloping side length $x$.
Find an expression, in terms of $x$, for the perimeter. Give your answer in its simplest form. ....................................................... [2]
596
545
513
A unit triangle is an equilateral triangle of side length 1.
There are 3 unit triangles in this trapezium with $x = 1$ and $y = 1$.
There are 12 unit triangles in this trapezium with $x = 2$ and $y = 2$.
(a) Work out the number of unit triangles in a trapezium with $x = 1$ and $y = 3$. You may use the grid below to help you. ................................................. [2]
(b) Complete the table. You may use the grid below to help you.
[Table_1]
| Shorter parallel side length (x) | Sloping side length (y) | Number of unit triangles |
|---------------------------------|-------------------------|--------------------------|
| 1 | 1 | 3 |
| 2 | 1 | |
| 3 | 1 | |
| 1 | 2 | |
| 2 | 2 | 12 |
| 3 | 2 | |
| 1 | 3 | |
| 2 | 3 | |
| 3 | 3 | 27 |
[3]
515
574
510
In a different sequence of trapeziums the shorter parallel side length, $x$, is equal to the sloping side length, $y$.
(a) The diagram shows the trapezium with $x = 4$ and $y = 4$.
Find the number of unit triangles in this trapezium.
............................................. [1]
(b) Complete the table.
You may use results from Questions 5(b) and 6(a), and patterns to help you.
[Table_1]
| Shorter parallel side length $(x)$ | Number of unit triangles |
|-----------------------------------|----------------------------------|
| 1 | $3 = 3 \times \text{..........} \quad = 3 \times 1^2$ |
| 2 | $\text{..........} = 3 \times 4 = 3 \times 2^2$ |
| 3 | $\text{..........} = 3 \times \text{..........} \quad = 3 \times \text{..........}$ |
| 4 | $\text{..........} = 3 \times \text{..........} \quad = 3 \times \text{..........}$ | [2]
(c) Write down an expression, in terms of $x$, for the number of unit triangles.
............................................. [1]
(d) In a trapezium the shorter parallel side length is equal to the sloping side length.
There are 675 unit triangles in this trapezium.
Find the perimeter of this trapezium.
............................................. [4]
525
562
528
