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INVESTIGATION EQUABLE SHAPES
In this investigation, lengths are given in centimetres.
The area of a shape is $A$ square centimetres and its perimeter is $P$ centimetres.
This task investigates the dimensions of equable shapes.
The shape is equable if $A = P$.
All the diagrams in this investigation are not to scale.
(a)
This rectangle is equable.
Write down the calculations to show that $A = 16.2$ and $P = 16.2$.
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(b) All the rectangles in this table are equable.
Complete the table.
[Table_1]
(c) For the rectangle in part (a), the value of $(x - 2)(y - 2)$ is
$$\begin{align} (4.5 - 2)(3.6 - 2) &= 2.5 \times 1.6 \\ &= 4.\end{align}$$
The rectangles in the table are equable.
Use your answers to part (b) to complete the first two columns of the table.
Calculate the value of $(x - 2)(y - 2)$ for each rectangle.
[Table_2]
(d) Use what you notice about the value of $(x - 2)(y - 2)$ in part (c) to find all the equable rectangles that have integer lengths and widths.
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578
517
583
The area, $A$, of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$.
(a) ![Triangle Image 1],
This right-angled triangle is equable. Write down the calculations to show that $A = 23.4$ and $P = 23.4$.
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(b) ![Triangle Image 2],
(i) Write down, in its simplest form, an expression, in terms of $x$, for the perimeter of this triangle.
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(ii) Write down, in its simplest form, an expression, in terms of $x$, for the area of this triangle.
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(iii) This triangle is equable. Using your answers to part (i) and part (ii) find $x$. Write down the length of each side.
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(c) ![Triangle Image 3],
In the diagram, $x$, $y$ and $z$ are the lengths of the sides of a right-angled triangle. All the right-angled triangles in the table below are equable. Use this fact and your answer to part (b)(iii) to complete the table.
| $x$ | $y$ | $z$ | Area ($A$) | Perimeter ($P$) |
|---|---|---|---|---|
| 6.5 | 7.2 | 9.7 | 23.4 | 23.4 |
| 20 | | | | 33.6 |
| 14 | 14.8 | | | |
| 5.6 | 9 | | | |
(d) For the triangle in part (a), the value of $(x - 4)(y - 4)$ is $(6.5 - 4)(7.2 - 4) = 2.5 \times 3.2 = 8.$
The triangles in the table are equable. Use your answers to part (c) to complete the first column of this table. Calculate the value of $(x - 4)(y - 4)$ for each triangle.
| $x$ | $y$ | $(x - 4)(y - 4)$ |
|---|---|---|
| 6.5 | 7.2 | 2.5 \times 3.2 = 8 |
| 4.4 | 24 | |
| 20 | | |
| 14 | | |
| 5.6 | 9 | |
(e) Use what you notice about the value of $(x - 4)(y - 4)$ in part (d) to find all the equable right-angled triangles that have integer bases and heights. Find the lengths of the three sides for each triangle.
569
600
560
