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Counter-intuitive triangle areas

The area of a triangle can be calculated as half of the length of its base multiplied by its perpendicular height. This means that two triangles with the same length base and the same perpendicular height have the same area.

Some people can find this counter-intuitive, but the areas can be proved to be identical by slicing up one triangle into several pieces, then rearranging the pieces to form the other triangle. This approach can be used to prove that the following two long, thin triangles have the same area:

[![Two triangles on a square grid, the second triangle one grid square wider than the first](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_8_9.svg)](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_8_9.svg)

The triangles are displayed on a grid to make it clear that both have a base of length 1 grid square (horizontally), and a perpendicular height of 1 grid square (vertically). The first triangle has a horizontal width of 8 grid squares, whereas the second triangle has a horizontal width of 9 grid squares.

## Puzzle
What is the smallest number of straight line cuts through the first triangle that will give pieces that can be rearranged into the second triangle?

The cuts can be in any location at any angle. They are not required to align with the grid squares.

## Hint
If you're not sure where to start, you can try the following (hidden to avoid spoilers for anyone who wants to try this unaided):

<details><summary>Hidden spoiler</summary>

You may find it easier to start with a smaller problem, and gradually work up until you see a pattern.

Here are the smaller versions of this puzzle in increasing size:

### Width 1 to width 2

[![Two triangles on a square grid, the second triangle one grid square wider than the first](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_1_2.svg)](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_1_2.svg)

### Width 2 to width 3

[![Two triangles on a square grid, the second triangle one grid square wider than the first](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_2_3.svg)](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_2_3.svg)

### Width 3 to width 4

[![Two triangles on a square grid, the second triangle one grid square wider than the first](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_3_4.svg)](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_3_4.svg)

### Width 4 to width 5

[![Two triangles on a square grid, the second triangle one grid square wider than the first](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_4_5.svg)](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_4_5.svg)

### Width 5 to width 6

[![Two triangles on a square grid, the second triangle one grid square wider than the first](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_5_6.svg)](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_5_6.svg)

### Width 6 to width 7

[![Two triangles on a square grid, the second triangle one grid square wider than the first](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_6_7.svg)](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_6_7.svg)

### Width 7 to width 8

[![Two triangles on a square grid, the second triangle one grid square wider than the first](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_7_8.svg)](https://raw.githubusercontent.com/trichoplax/triangle-diagrams/refs/heads/main/width_7_8.svg)

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