Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S, is _______.Correct answer is '36'. Can you explain this answer?

Question

Answer

But all cases are possible, so total number of positive cases = 9 + 9 + 9 + 9 = 36

Answer

Understanding the Triangle Area Formula
To find how many triangles have an area of 50 square units with vertices at the origin and on the coordinate axes, we start with the area formula for a triangle:
- The area A of a triangle with base b and height h is given by:
A = (1/2) * b * h
Setting Up the Equation
Given that the area A is 50 square units, we can write:
- 50 = (1/2) * b * h
Multiplying both sides by 2 gives:
- b * h = 100
Identifying Integral Coordinates
Since the vertices are on the coordinate axes, we denote:
- One vertex at (b, 0) on the x-axis
- Another vertex at (0, h) on the y-axis
We need to find all pairs of positive integers (b, h) such that:
- b * h = 100
Finding Factor Pairs
The factors of 100 are crucial here. The positive integer factor pairs (b, h) that satisfy this equation are:
- (1, 100)
- (2, 50)
- (4, 25)
- (5, 20)
- (10, 10)
- (20, 5)
- (25, 4)
- (50, 2)
- (100, 1)
This gives us a total of 9 pairs.
Considering Quadrants
Each pair (b, h) can create a triangle in four different quadrants because we can have:
- (b, h)
- (-b, h)
- (b, -h)
- (-b, -h)
Thus, for each of the 9 pairs, we can create 4 triangles.
Calculating the Total Triangles
To find the total number of triangles, we multiply the number of pairs by the number of quadrants:
- Total triangles = 9 pairs * 4 quadrants = 36 triangles
Hence, the final answer for the number of elements in the set S is 36.

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