A triangle is formed with vertices (0, 0), (0, 100) and (100, 100). What is the number of points inside the triangle with integer coordinates?
[amp_mcq option1=”5000″ option2=”4999″ option3=”4851″ option4=”4800″ correct=”option3″]
The lines forming the boundaries of the triangle are:
1. The line segment AB is along the y-axis, from y=0 to y=100 (x=0).
2. The line segment BC is along the line y=100, from x=0 to x=100.
3. The line segment AC connects (0, 0) and (100, 100). The equation of this line is y = x.
A point (x, y) is strictly inside this triangle if it satisfies the following conditions:
1. It must be to the right of the line x=0: x > 0.
2. It must be below the line y=100: y < 100. 3. It must be above the line y=x: y > x.
Combining these, we are looking for integer coordinates (x, y) such that 0 < x < y < 100. Let's iterate through possible integer values for x. Since x > 0 and x < y < 100, the smallest possible integer value for x is 1. If x = 1, y must be an integer such that 1 < y < 100. Possible y values are 2, 3, ..., 99. The number of such y values is 99 - 2 + 1 = 98. If x = 2, y must be an integer such that 2 < y < 100. Possible y values are 3, 4, ..., 99. The number of such y values is 99 - 3 + 1 = 97. If x = 3, y must be an integer such that 3 < y < 100. Possible y values are 4, 5, ..., 99. The number of such y values is 99 - 4 + 1 = 96. ... What is the largest possible integer value for x? Since x < y < 100, the largest possible integer value for y is 99. This requires x to be at least 1 less than 99, i.e., x < 99. So, the largest possible integer value for x is 98. If x = 98, y must be an integer such that 98 < y < 100. The only possible y value is 99. The number of such y values is 1. (99 - 99 + 1 = 1) The total number of integer points inside the triangle is the sum of the number of possible y values for each x from 1 to 98. Total points = (99 - 1) + (99 - 2) + (99 - 3) + ... + (99 - 98) Total points = 98 + 97 + 96 + ... + 1 This is the sum of the first 98 positive integers. The formula for the sum of the first n positive integers is n(n+1)/2. Here, n = 98. Sum = 98 * (98 + 1) / 2 = 98 * 99 / 2 = 49 * 99. 49 * 99 = 49 * (100 - 1) = 4900 - 49 = 4851. The number of points inside the triangle with integer coordinates is 4851.