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- In a plane, there are fourteen points. Of these fourteen points, five points are in a straight
In a plane, there are fourteen points. Of these fourteen points, five points are in a straight line, and except for these five points, no other three points are in the same straight line. Find: (i) The number of straight lines formed. (ii) The number of rectangles formed. (iii) The number of hexagons formed by joining these fourteen points.
(i) 32, (ii) 36, (iii) 9
(i) 37, (ii) 32, (iii) 9
(i) 37, (ii) 36, (iii) 8
(i) 37, (ii) 36, (iii) 9
Answers (1)
(i) The number of straight lines formed:
To find the number of straight lines formed, we need to consider the five points that are in a straight line and count the additional lines that can be formed by connecting the remaining nine points to each other and to the five points on the straight line.
The number of additional lines that can be formed by connecting the remaining nine points to each other is given by the combination formula, which is .
Therefore, the total number of straight lines formed is 1 (the straight line formed by the five points) + 36 (additional lines formed by connecting the remaining nine points) = 37 .
(ii) The number of rectangles formed:
To form a rectangle, we need to consider the five points on the straight line as the opposite vertices of the rectangle and choose two more points from the remaining nine points as the other two vertices.
The number of ways to choose two points from the remaining nine points is given by the combination formula, which is .
Therefore, the number of rectangles formed is .
(iii) The number of hexagons formed:
To form a hexagon, we need to consider the five points on the straight line as consecutive vertices of the hexagon and choose one more point from the remaining nine points as the sixth vertex.
The number of ways to choose one point from the remaining nine points is given by the combination formula, which is .
Therefore, the number of hexagons formed is 9.
So, the answers are:
(i) The number of straight lines formed: 37
(ii) The number of rectangles formed: 36
(iii) The number of hexagons formed: 9
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