Question
Previous Question
Next Question
Question 3
Using diagonals from a common vertex, how many triangles could be formed from a decagon?
Answer
Submit Answer
Copyright ©2025 DeltaMath.com All Rights Reserved.
Terms & Policies
Question 3
Submit Answer
Copyright ©2025 DeltaMath.com All Rights Reserved.
Terms & Policies
Ask by Pope Dickson. in the United States
Jan 16,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
21 triangles can be formed using diagonals from a common vertex in a decagon.
Solution
To determine how many triangles can be formed using diagonals from a common vertex in a decagon (a 10-sided polygon), let’s break down the problem step by step.
Step 1: Understanding the Decagon
A decagon has 10 vertices. For any given vertex:
- It is connected to 2 adjacent vertices via the sides of the decagon.
- It can be connected to the remaining 7 vertices via diagonals.
Step 2: Choosing the Common Vertex
Let’s select one vertex as the common vertex (let’s call it A) from which all diagonals will be drawn.
Step 3: Drawing Diagonals from Vertex A
From vertex A, you can draw diagonals to 7 other vertices (since connections to the adjacent vertices are sides, not diagonals).
Step 4: Forming Triangles
To form a triangle using these diagonals:
-
Select any two diagonals from the 7 available. Each pair of diagonals will form a triangle with vertex A.
-
The number of ways to choose 2 diagonals from 7 is given by the combination formula:
-
Calculating this:
Conclusion
21 triangles can be formed using diagonals from a common vertex in a decagon.
Answer: 21
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Extra Insights
When dealing with a decagon, which has 10 sides, you can use the formula for finding the number of triangles that can be formed by joining diagonals from a common vertex. Specifically, if you pick one vertex, you can connect it to other vertices, excluding itself and its two adjacent vertices. So, you would choose from 7 additional vertices, which gives you
To visualize this, imagine standing at one corner of the decagon. You can draw lines to several other corners, creating a web of triangles. Each time you choose two other vertices to connect with the original vertex, you’re forming a new triangle. It’s like a party where you can invite different combinations of friends to join you for a thrilling geometry dance!
