If all of the diagonals are drawn from a vertex of a pentagon, how many triangles are formed?
To determine the number of triangles formed when all diagonals are drawn from a vertex of a pentagon, we need to understand the pattern and reasoning behind it.
Let's start by considering the number of diagonals that can be drawn from a single vertex of a pentagon. From any given vertex, we can draw diagonals to the other four vertices of the pentagon, resulting in four diagonals.
Now, let's focus on the process of forming triangles using these diagonals. To form a triangle, we need three vertices, and the diagonals will serve as the connecting lines between these vertices. Since we have five vertices in total, we can select three of them to form a triangle.
To calculate the number of triangles formed, we can use the combination formula. In this case, we can express it as "5 choose 3" or written as C(5, 3). The combination formula is given by the formula:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of elements, and r is the number of elements chosen. Using this formula, we can calculate:
C(5, 3) = 5! / (3! * (5 - 3)!)
= 5! / (3! * 2!)
= (5 * 4 * 3!) / (3! * 2 * 1)
= 5 * 4 / (2 * 1)
= 10
Therefore, when all diagonals are drawn from a vertex of a pentagon, a total of 10 triangles are formed.
If you would like to verify this, you can manually draw a pentagon and draw all the diagonals from a single vertex. Counting the number of triangles formed will reinforce the result obtained through the combination formula.