\nThe vertices of the triangle are A=(0, 0), B=(0, 100), and C=(100, 100).
\nThe lines forming the boundaries of the triangle are:
\n1. The line segment AB is along the y-axis, from y=0 to y=100 (x=0).
\n2. The line segment BC is along the line y=100, from x=0 to x=100.
\n3. The line segment AC connects (0, 0) and (100, 100). The equation of this line is y = x.<\/p>\nA point (x, y) is strictly inside this triangle if it satisfies the following conditions:
\n1. It must be to the right of the line x=0: x > 0.
\n2. It must be below the line y=100: y < 100.\n3. It must be above the line y=x: y > x.<\/p>\n
Combining these, we are looking for integer coordinates (x, y) such that 0 < x < y < 100.\n\nLet's iterate through possible integer values for x.\nSince x > 0 and x < y < 100, the smallest possible integer value for x is 1.\nIf 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.\nIf 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.\nIf 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.\n...\nWhat 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.\nIf 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)\n\nThe 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.\nTotal points = (99 - 1) + (99 - 2) + (99 - 3) + ... + (99 - 98)\nTotal points = 98 + 97 + 96 + ... + 1\n\nThis 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.\nHere, n = 98.\nSum = 98 * (98 + 1) \/ 2 = 98 * 99 \/ 2 = 49 * 99.\n49 * 99 = 49 * (100 - 1) = 4900 - 49 = 4851.\n\nThe number of points inside the triangle with integer coordinates is 4851.\n<\/section>\n","protected":false},"excerpt":{"rendered":"
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″] This question was previously asked in UPSC CAPF – 2016 Download PDFAttempt Online The correct option is C) 4851. The problem asks for the … <\/p>\n
Detailed SolutionA triangle is formed with vertices (0, 0), (0, 100) and (100, 100). Wh<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1098,1102],"class_list":["post-90015","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1098","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nA triangle is formed with vertices (0, 0), (0, 100) and (100, 100). Wh<\/title>\n<meta name=\"description\" content=\"The correct option is C) 4851. 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