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Given a cube, I pick three random points from among its vertices. Let N be the number of ways in which three points can be selected such that they form a right angled triangle. What is N?
Correct answer is '48'. Can you explain this answer?
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Given a cube, I pick three random points from among its vertices. Let ...
The number of ways in which three points can be selected out of the eight vertices of a cube is 8C3 = 56
We can see that in most of these 56 ways, the three points selected form a right angled triangle. So, we look at possible selections of three points which would not give a right angled triangle.
We can see that if two of these three points are the endpoints of one of the edges of the cube, then irrespective of which of the remaining vertices is the third point, we always get a right angle. If two of the three points are endpoints of a body diagonal of the cube, then the third point will always be such that it will form an edge of the cube with one of the two points, so we again get a right angle.
So, the sides of the triangle formed by the three points should neither be edges nor the body diagonals of the cube, i.e. all three sides should be face diagonals.
To count the number of such triangles, consider the 'top' face ABCD of a cube ABCDWXYZ (the points corresponding to A, B, C and D on the bottom face are W, X, Y and Z respectively). It has two face diagonals, AC and BD. With AC as one of the sides, we can have two acute triangles, ΔACX and ΔACZ. With BD as one of the sides, we can have two acute triangles, ΔBDY and ΔBDW.
Similarly, WXYZ has two face diagonals, WY and XZ, from which we get four acute triangles, ΔBWY, ΔDWY, ΔAXZ and ΔCXZ. Thus, 8 triangles out of 56 are not right angled triangles.
56 - 8 = 48
Answer: 48
We can see that in most of these 56 ways, the three points selected form a right angled triangle. So, we look at possible selections of three points which would not give a right angled triangle.
We can see that if two of these three points are the endpoints of one of the edges of the cube, then irrespective of which of the remaining vertices is the third point, we always get a right angle. If two of the three points are endpoints of a body diagonal of the cube, then the third point will always be such that it will form an edge of the cube with one of the two points, so we again get a right angle.
So, the sides of the triangle formed by the three points should neither be edges nor the body diagonals of the cube, i.e. all three sides should be face diagonals.
To count the number of such triangles, consider the 'top' face ABCD of a cube ABCDWXYZ (the points corresponding to A, B, C and D on the bottom face are W, X, Y and Z respectively). It has two face diagonals, AC and BD. With AC as one of the sides, we can have two acute triangles, ΔACX and ΔACZ. With BD as one of the sides, we can have two acute triangles, ΔBDY and ΔBDW.
Similarly, WXYZ has two face diagonals, WY and XZ, from which we get four acute triangles, ΔBWY, ΔDWY, ΔAXZ and ΔCXZ. Thus, 8 triangles out of 56 are not right angled triangles.
56 - 8 = 48
Answer: 48
Most Upvoted Answer
Given a cube, I pick three random points from among its vertices. Let ...
There are 8 vertices, 3 can be selected in 8C3 ways which is 56. Number ways when it doesn't become a right angle is when we choose from the same plane so 2*4C2 so it is 8. Required is 56-8=48
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Given a cube, I pick three random points from among its vertices. Let ...
Introduction:
In this problem, we are given a cube and we need to select three random points from its vertices such that they form a right-angled triangle. We need to determine the number of ways to select these points.
Approach:
To solve this problem, we can consider the different types of right-angled triangles that can be formed by selecting three points on the cube. We can then count the number of ways to select each type of triangle and add them up to get the total number of ways.
Types of right-angled triangles:
1. Triangular face: A right-angled triangle can be formed by selecting three vertices on one of the six triangular faces of the cube. There are six such faces, so the number of ways to select a triangle on a triangular face is 6.
2. Rectangular face: A right-angled triangle can also be formed by selecting two vertices on one of the twelve rectangular faces of the cube and one vertex on an adjacent rectangular face. There are twelve pairs of adjacent rectangular faces, so the number of ways to select a triangle on a rectangular face is 12.
3. Opposite vertices: A right-angled triangle can be formed by selecting two opposite vertices of the cube and one vertex on a face adjacent to one of the opposite vertices. There are eight pairs of opposite vertices, so the number of ways to select a triangle using opposite vertices is 8.
Calculating the total:
To calculate the total number of ways, we need to sum up the number of ways for each type of right-angled triangle:
Total number of ways = Number of ways on triangular faces + Number of ways on rectangular faces + Number of ways using opposite vertices
Total number of ways = 6 + 12 + 8 = 26
Answer:
Therefore, the total number of ways to select three points on the cube such that they form a right-angled triangle is 26.
In this problem, we are given a cube and we need to select three random points from its vertices such that they form a right-angled triangle. We need to determine the number of ways to select these points.
Approach:
To solve this problem, we can consider the different types of right-angled triangles that can be formed by selecting three points on the cube. We can then count the number of ways to select each type of triangle and add them up to get the total number of ways.
Types of right-angled triangles:
1. Triangular face: A right-angled triangle can be formed by selecting three vertices on one of the six triangular faces of the cube. There are six such faces, so the number of ways to select a triangle on a triangular face is 6.
2. Rectangular face: A right-angled triangle can also be formed by selecting two vertices on one of the twelve rectangular faces of the cube and one vertex on an adjacent rectangular face. There are twelve pairs of adjacent rectangular faces, so the number of ways to select a triangle on a rectangular face is 12.
3. Opposite vertices: A right-angled triangle can be formed by selecting two opposite vertices of the cube and one vertex on a face adjacent to one of the opposite vertices. There are eight pairs of opposite vertices, so the number of ways to select a triangle using opposite vertices is 8.
Calculating the total:
To calculate the total number of ways, we need to sum up the number of ways for each type of right-angled triangle:
Total number of ways = Number of ways on triangular faces + Number of ways on rectangular faces + Number of ways using opposite vertices
Total number of ways = 6 + 12 + 8 = 26
Answer:
Therefore, the total number of ways to select three points on the cube such that they form a right-angled triangle is 26.
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Given a cube, I pick three random points from among its vertices. Let N bethe number of ways in which three points can be selected such that they form a right angled triangle. What is N?Correct answer is '48'. Can you explain this answer?
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