The number of diagonals in a polygon = n(n-3)/2, where n is the number of polygon sides.
For a convex n-sided polygon, there are n vertexes, and from each vertex you can draw n-3 diagonals, so the total number of diagonals that can be drawn is n (n-3).
However, this would mean that each diagonal would be drawn twice, (to and from each vertex), so the expression must be divided by 2.
nC2 (choose 2) - n sides = n(n-1)/ 2 – n sides
2. Example: There are n people at a party, each person shakes hands with the every other person once. How many handshakes?Method I:nC2 in this case (10 x 9) /2 =45
__ __ First slot you have 10 persons to choose from, second slot 9 persons. Since A shakes hands with B is the same as B shakes hands with A, so you divide the number by 2 and get the answer.
Method II:Sum of the first consecutive Natural numbers: n (n+1) /2
The first person shakes hands with 9 other person; the second person shakes hands with 8 other person, etc…
So 9 + 8 + 7 + …= (9 x 10)/ 2 = 45
3. Example: N dots evenly spaced on a circle. How many chords can you make using those dots?
Methods: This is very similar to hand-shaking questions.
II: Sum of the first consecutive (N-1) Natural numbers
Word problems: Answers below.
#1 : A convex polygon with n sides has 20 diagonals. How many diagonals does an (n+1)-sided convex polygon have?
#2: A polygon has n sides and n diagonals. What is n?
#3: How many diagonals does a decagon have?
#4: How many diagonals does a dodecagon have?
#5: How many line segments have both their endpoints located at the vertexes of a given cube?
#6: There are 8 points on a circle, how many lines can you make? How many triangles can you make?
#7: If each of the interior angles of two regular polygons adds up to 195 degrees and their diagonals add up to 29, what is the sum of their sides?
#8: A regular polygon (see left image) is covered by a piece of paper. If the two lower angles are each 36 degrees. How many sides would this regular polygon have?
# 1- 27 #2 -5 #3-35 #4-54 [12C2 -12] #5 28 [There are 8 vertices, so 8C2]
#6 8C2 = 28 for lines and 8C3 = 56 triangles
#7: 6[hexagon] + 8[octagon] = 14 sides
#8: 10 sides - Since the shown part is a quadrilateral polygon, it has 360 degrees total.
[360 -(36 x 2)] / 2 = 144; 180 - 360/x = 144; x =10 [The exterior angles of any polygon add up to 360 degrees so 360/ sides will give you one exterior angle of a regular polygon and 180 - exterior angle of a
regular polygon will give you one interior angle of a regular polygon.