How many triangles are formed by drawing all the diagonals from a single vertex?
Understand the Problem
The question is asking us to calculate how many triangles can be formed by drawing all the diagonals from a single vertex in a convex polygon, specifically an octagon in this case.
Answer
$10$
Answer for screen readers
The number of triangles formed is $10$.
Steps to Solve
- Identify the number of sides in the polygon
An octagon has 8 sides.
- Determine the vertex to draw diagonals from
Choose one vertex in the octagon. In this case, let's say we choose vertex A.
- Count the vertices you can connect with diagonals
From vertex A, you can connect to other vertices except for itself and the two adjacent vertices. In total, this means: $$ 8 - 1 - 2 = 5 $$ So you can draw 5 diagonals from vertex A.
- Form triangles with the diagonals and the chosen vertex
Each diagonal connects vertex A with another vertex that is not adjacent. To form a triangle, you need to choose 2 out of the 5 non-adjacent vertices: $$ \text{Number of triangles} = \binom{5}{2} $$ Where $\binom{n}{r}$ is the binomial coefficient used to calculate combinations.
- Calculate the binomial coefficient
Now, calculate $\binom{5}{2}$: $$ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 $$
The number of triangles formed is $10$.
More Information
In an octagon, drawing all the diagonals from a single vertex creates several triangles. The reasoning applies to any convex polygon where you start from a vertex and form triangles with non-adjacent vertices.
Tips
- Forgetting to subtract the vertex itself and the two adjacent vertices when counting possible diagonals from a vertex.
- Confusing the binomial coefficient calculation; ensure you understand how to compute combinations.
AI-generated content may contain errors. Please verify critical information
