The following sequence of figures begins repeating in the fifth figure.
The fifth figure has 2 squares, 2 triangles, and 1 rhombus. How many triangles will there be in the 176th figure?
To find the number of triangles in the 176th figure, we first need to understand the pattern of how the number of triangles changes as we move from one figure to the next.
From the information given, we know that the fifth figure has 2 triangles. Let's analyze the pattern:
In the fifth figure:
- Number of squares = 2
- Number of triangles = 2
- Number of rhombuses = 1
Notice that the number of squares, triangles, and rhombuses in each figure is the same. This suggests that the repeating pattern is based on these three shapes.
To find the number of triangles in the 176th figure, we need to determine where in the repeating pattern the 176th figure falls.
Since the pattern repeats after the fifth figure, we can use modular arithmetic to find the position of the 176th figure in the pattern.
176 modulo 3 = 2
The remainder 2 tells us that the 176th figure is two positions after the fifth figure in the pattern. Therefore, we need to look at the second position in the pattern to determine the number of triangles.
In the second position:
- Number of squares = 2
- Number of triangles = 2
- Number of rhombuses = 1
Therefore, the 176th figure will also have 2 triangles.
So, the answer is: There will be 2 triangles in the 176th figure.