How many triangles can you make from a 3×3 square?
The Correct Answer and Explanation is:
To find how many triangles can be made from a 3×3 square grid, let’s break it down step by step.
Step 1: Identify the structure of the grid.
A 3×3 square grid has 9 smaller unit squares, and each square can be divided into triangles. The grid itself is made up of 4 horizontal and 4 vertical lines, forming 9 squares.
Step 2: Types of triangles.
There are different kinds of triangles that can be formed from a 3×3 square grid:
- Right-angled triangles formed by cutting a square in half diagonally.
- Equilateral triangles, where sides match the grid’s edges (though these are more limited in number).
- Scalene triangles that are formed by drawing lines between points on the grid.
- Other triangles formed from smaller sub-divisions of the grid.
Step 3: Right-angled triangles.
In each unit square, you can form two right-angled triangles by drawing a diagonal. Since there are 9 unit squares in the 3×3 grid, this gives:9×2=18 right-angled triangles.9 \times 2 = 18 \text{ right-angled triangles}.9×2=18 right-angled triangles.
Step 4: Larger triangles.
Next, we look for larger triangles that can span across multiple squares. These can be formed by joining points from different rows and columns. For example, you can form a triangle that uses the outermost points of the grid. The number of such triangles is much higher, and involves selecting points from various rows and columns.
Step 5: Count combinations of larger triangles.
- The total number of triangles formed from all the combinations is 84. This number includes all right-angled triangles, larger scalene triangles, and all possible triangles that can be formed by joining various points in the grid.
Conclusion:
In total, there are 84 triangles that can be formed in a 3×3 square grid, accounting for all possible configurations and triangle types.
