Match the polygon and line-counting results. Column I: (A) A convex n -gon has 210 distinct interior intersection points of diagonals: find n . (B) A convex polygon has equally many sides and diagonal

Question

Match the polygon and line-counting results. Column I: (A) A convex $n$ -gon has 210 distinct interior intersection points of diagonals: find $n$ . (B) A convex polygon has equally many sides and diagonals: find its sides. (C) An even-sided regular polygon has 126 diagonals not passing through its circumcircle center: find its sides. (D) $n$ plane lines, with none parallel and no triple concurrency, divide the plane into 172 regions: find $n$ . Column II: (p) 18 (q) 16 (r) 10 (s) 5

Column I	Column II
(A) A convex $n$ -gon has 210 distinct interior intersection points of diagonals: find $n$ .	(p) 18
(B) A convex polygon has equally many sides and diagonals: find its sides.	(q) 16
(C) An even-sided regular polygon has 126 diagonals not passing through its circumcircle center: find its sides.	(r) 10
(D) $n$ plane lines, with none parallel and no triple concurrency, divide the plane into 172 regions: find $n$ .	(s) 5

TestHub · Accepted Answer

( 4 n ​ ) = 210 gives ten only if each quadruple gives a distinct intersection. In general concurrency merges points. B: n = n ( n − 3 ) /2 gives five. C: n ( n − 3 ) /2 − n /2 = 126 gives 18. D: 1 + n ( n + 1 ) /2 = 172 gives 18. Note: A assumes general position (no three diagonals concurrent).

Mathematics - Permutation & Combination Question with Solution | TestHub

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