How many triangles are present in the given figure?
1. Identify the Sets of Lines
The figure is composed of three types of lines:
- Horizontal-ish lines ($H$): There are 5 lines (the bottom base, 3 internal lines, and the top base).
- Vertical-ish lines ($V$): There are 5 lines (the left side, 3 internal lines, and the right side).
- Diagonal line ($D$): A single line connecting the bottom-left vertex to the top-right vertex.
2. Understand Triangle Formation
For a triangle to be formed, we need three non-parallel lines that do not all intersect at the same point.
- Since all lines in set $H$ are parallel to each other, no two lines from $H$ can form a triangle.
- Similarly, no two lines from set $V$ can form a triangle.
- Therefore, every triangle in this figure must include the diagonal line ($D$) as one of its sides.
- To form a triangle with the diagonal, we must choose one line from set $H$ and one line from set $V$ that intersect the diagonal at two different points.
3. Mathematical Calculation
Each intersection of a horizontal line and a vertical line (a "grid point") represents a potential triangle vertex when paired with the diagonal. There are:
$ 5 \text{ (horizontal lines)} \times 5 \text{ (vertical lines)} = 25 \text{ intersection points} $
However, if a horizontal line, a vertical line, and the diagonal line all meet at the same point, they do not form a triangle (the area is zero). In a $4 \times 4$ grid (5 lines by 5 lines) where the diagonal connects opposite corners, the diagonal passes exactly through the grid points where the indices are equal (e.g., $(0,0), (1,1), (2,2), (3,3), (4,4)$).
- Total intersections: 25
- Points where the diagonal meets a grid intersection: 5
The number of triangles is the total number of intersections minus the number of points where the lines are concurrent:
$ \text{Total Triangles} = 25 - 5 = 20 $
Important Questions from 2D 3D Pattern Visualization
Three different views of a dice are shown in the figure below.
The piece of paper that can be folded to make this dice is
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From the additional plates given in the options, which one of the combinations of additional plates would allow the player to construct a five-letter palindrome. The player should use all the five plates exactly once. The plates can be rotated in their plane.
