What is the total number of triangles in the figure given below?
Show Hint
In complex figure counting problems, start by counting the smallest, most obvious shapes first. Then, systematically look for larger shapes formed by combining the smaller ones. It can be helpful to redraw the figure and trace over each counted shape to avoid errors.
Step 1: Understanding the Question:
The task is to count all the triangles present in the given complex line drawing of an animal figure. Step 2: Key Formula or Approach:
The most effective method for such problems is to break the figure into smaller, distinct sections, count the triangles in each section, and then look for larger triangles formed by combining these sections. This systematic approach helps prevent miscounting. Step 3: Detailed Explanation:
Let's count the triangles by dividing the figure into five main regions: 1. Head Section (Right Part):
- The top 'horn' part is a triangle divided by one line, creating 3 triangles (2 small + 1 large).
- The central 'face' part is a quadrilateral with its two diagonals drawn, which always forms 8 triangles (4 small inner triangles + 4 larger triangles formed by combining adjacent pairs).
- The bottom 'jaw' part is also a triangle divided by one line, creating another 3 triangles.
- Total for Head = \(3 + 8 + 3 = 14\) triangles. 2. Front Legs Section (Leftmost Part):
- This section is a large triangle divided by a line from a vertex to the opposite side, creating 3 triangles (2 smaller + 1 large). 3. Hind Legs Section (Behind the Head):
- This section is identical in construction to the front legs, so it also contains 3 triangles. 4. Body Section (Central Circle Area):
- There is a large triangle inscribed in the circle. A point inside this triangle is connected to its three vertices, dividing it into three smaller triangles. This gives a total of 4 triangles (3 small + 1 large outer). 5. Large Connecting Triangles:
- One large triangle is formed by the line of the animal's back, connecting the head and body. (1 triangle)
- Another large triangle is formed by the belly line, also connecting the head and body. (1 triangle)
- A final, often missed, large triangle is formed by the top vertex of the body's inscribed triangle, the rightmost point of the circle, and the bottom-most point of the jaw section. (1 triangle) Total Count:
Summing the triangles from all sections:
\[ \text{Total} = (\text{Head}) + (\text{Front Legs}) + (\text{Hind Legs}) + (\text{Body}) + (\text{Connecting}) \]
\[ \text{Total} = 14 + 3 + 3 + 4 + (1 + 1 + 1) = 27 \text{ triangles} \]
Step 4: Final Answer:
The total number of triangles in the figure is 27. This corresponds to option (D).
