In a convex octagon match triangle and diagonal counts. Column I: (A) Triangles with exactly one side an octagon side (B) Triangles with exactly two sides octagon sides (C) Triangles with no side an o

Question

In a convex octagon match triangle and diagonal counts. Column I: (A) Triangles with exactly one side an octagon side (B) Triangles with exactly two sides octagon sides (C) Triangles with no side an octagon side (D) Diagonals of the octagon Column II: (p) 16 (q) 7 (r) 32 (s) 20 (t) 8

Column I	Column II
(A) Triangles with exactly one side an octagon side	(p) 16
(B) Triangles with exactly two sides octagon sides	(q) 7
(C) Triangles with no side an octagon side	(r) 32
(D) Diagonals of the octagon	(s) 20
	(t) 8

TestHub · Accepted Answer

There are ( 3 8 ​ ) = 56 triangles. Exactly two boundary sides means three consecutive vertices (8 choices). Exactly one gives 8 ( 8 − 4 ) = 32 choices, leaving 56 − 8 − 32 = 16 with none. The diagonals number 8 ( 8 − 3 ) /2 = 20 .

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