Six points lie on the circumference of a circle. How many different triangles can be drawn having some of these points as vertices?
To determine the number of different triangles that can be drawn using the six points on the circumference of a circle, we need to consider the different possibilities.
1. Choose any 3 points:
There are 6 points on the circumference of the circle, so we can choose any 3 points out of the 6. We can calculate this using combinations, denoted as "nCr," where n is the total number of objects and r is the number of objects chosen.
In this case, we need to find the number of ways to choose 3 points out of 6, so the calculation is:
6C3 = (6!)/(3!(6-3)!) = (6x5x4)/(3x2) = 20
So, there are 20 different triangles that can be formed by choosing any 3 points.
2. Choose 3 consecutive points:
We can also consider choosing 3 consecutive points on the circumference of the circle. Since the circle is a closed curve, these points will always make a triangle. We can start choosing any of the 6 points and count the number of possibilities.
There are 6 consecutive sets of 3 points on a circle:
- Set 1: (1, 2, 3)
- Set 2: (2, 3, 4)
- Set 3: (3, 4, 5)
- Set 4: (4, 5, 6)
- Set 5: (5, 6, 1)
- Set 6: (6, 1, 2)
So, there are 6 different triangles that can be formed by choosing 3 consecutive points.
3. Choose 2 points and a midpoint:
In this case, we choose 2 points and a third point from the remaining 4 points to create a triangle. Again, we can use combinations to calculate the number of possibilities.
We have 6 points to choose from to form the first vertex of the triangle. Once we choose one point, there are 5 remaining points to choose from for the second vertex. Finally, out of the remaining 4 points, we choose one as the third midpoint. So, the calculation is:
6C1 * 5C1 * 4C1 = 6 * 5 * 4 = 120
So, there are 120 different triangles that can be formed by choosing 2 points and a midpoint.
Overall, the total number of different triangles that can be drawn using the six points on the circumference of a circle is:
Number of triangles formed by choosing any 3 points + Number of triangles formed by choosing 3 consecutive points + Number of triangles formed by choosing 2 points and a midpoint
= 20 + 6 + 120
= 146
Therefore, there are 146 different triangles that can be drawn using these six points.
To find out the number of different triangles that can be formed using six points on the circumference of a circle, we can use the concept of combinations.
We can choose any three points from the six points to form a triangle, as long as the three points do not lie on a straight line.
To calculate the number of combinations, we can use the formula for combinations:
C(n, r) = n! / (r!(n-r)!)
In this case, we have n = 6 (the total number of points) and r = 3 (the number of points needed to form a triangle).
Using the formula, we can calculate the number of combinations:
C(6, 3) = 6! / (3!(6-3)!)
= 6! / (3! * 3!)
= (6 * 5 * 4) / (3 * 2 * 1)
= 20
Therefore, there are 20 different triangles that can be formed using six points on the circumference of a circle.