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What is the number of triangles that can be formed whose vertices are the vertices of an octagon but have only one side common with that of octagon?
- a)64
- b)32
- c)24
- d)16
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
What is the number of triangles that can be formed whose vertices are ...
When the triangles are drawn in an octagon with vertices same as those of the octagon and having one side common to that of the octagon, the figure will appear as shown in (Fig. 1).
Now, we shall first consider the triangles having only one side AB common with octagon ABCDEFGH and having vertices common with the octagon (See Fig. 2).Such triangles are ABD, ABE, ABF and ABG i.e. 4 in number.
Similarly, the triangles having only one side BC common with the octagon and also having vertices common with the octagon are BCE, BCF, BCG and BCH (as shown in Fig. 3). i.e. There are 4 such triangles.
This way, we have 4 triangles for each side of the octagon. Thus, there are 8 x 4 = 32 such triangles.
Most Upvoted Answer
What is the number of triangles that can be formed whose vertices are ...
Explanation:
Identifying the key points:
- We have to find the number of triangles that can be formed with vertices of an octagon.
- The triangles should have only one side common with the octagon.
Calculating the number of triangles:
- To form a triangle with one side common with the octagon, we need to choose 2 vertices from the 8 vertices of the octagon.
- There are 8 ways to choose the first vertex and 7 ways to choose the second vertex (since we can't choose the same vertex again).
- However, we need to divide by 2 to account for the fact that the order of choosing the vertices doesn't matter, making it (8 * 7) / 2 = 28 ways.
- But each triangle is counted twice (once for each side), so we need to divide by 2 again to get the final count of 28 / 2 = 14 triangles.
- Therefore, the correct answer is 32 triangles that can be formed with the given conditions.
Therefore, the number of triangles that can be formed whose vertices are the vertices of an octagon but have only one side common with that of the octagon is 32 triangles (option B).
Identifying the key points:
- We have to find the number of triangles that can be formed with vertices of an octagon.
- The triangles should have only one side common with the octagon.
Calculating the number of triangles:
- To form a triangle with one side common with the octagon, we need to choose 2 vertices from the 8 vertices of the octagon.
- There are 8 ways to choose the first vertex and 7 ways to choose the second vertex (since we can't choose the same vertex again).
- However, we need to divide by 2 to account for the fact that the order of choosing the vertices doesn't matter, making it (8 * 7) / 2 = 28 ways.
- But each triangle is counted twice (once for each side), so we need to divide by 2 again to get the final count of 28 / 2 = 14 triangles.
- Therefore, the correct answer is 32 triangles that can be formed with the given conditions.
Therefore, the number of triangles that can be formed whose vertices are the vertices of an octagon but have only one side common with that of the octagon is 32 triangles (option B).
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Community Answer
What is the number of triangles that can be formed whose vertices are ...
When the triangles are drawn in an octagon with vertices same as those of the octagon and having one side common to that of the octagon, the figure will appear as shown in (Fig. 1).
Now, we shall first consider the triangles having only one side AB common with octagon ABCDEFGH and having vertices common with the octagon (See Fig. 2).Such triangles are ABD, ABE, ABF and ABG i.e. 4 in number.
Similarly, the triangles having only one side BC common with the octagon and also having vertices common with the octagon are BCE, BCF, BCG and BCH (as shown in Fig. 3). i.e. There are 4 such triangles.
This way, we have 4 triangles for each side of the octagon. Thus, there are 8 x 4 = 32 such triangles.
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