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The total number of interior angles in two regular polygons is 17, and the total number of diagnals is 53. How many sides does each regular polygon have?

2007-01-28 12:39:27 · 1 answers · asked by RIMAS 2 in Education & Reference Homework Help

1 answers

Step 1: Write out how the # of interior angles and # of diagonals relate to the # of sides.
# of interior angles = # of sides.
# of diagonals = (n-3) + ((n-3) * (n-2)/2)
Proof: For an octagon, the first point has 5 diagonals (8 points, -1 for itself, -2 for the points connected by a side). The next point has 5 diagonals, then each point thereafter has 1 less diagonal, because you can't double count previously counted diagonals. An octogon therefore has 5 + 5 + 4 + 3 + 2 + 1 = 20 diagonals. Another way to write 1 + 2 + 3 ... + n is (n * (n + 1)/2).

Step 2: Write these formulas using x and y for the 2 polygons:
# of interior angles = x + y = 17
# of diagonals = (x-3) + ((x-3) * (x-2)/2) + (y-3) + ((y-3) * (y-2)/2) = 53

Step 3: Use the # of interior angles to find x defined in terms of y:
x + y = 17
x = 17 - y

Step 4: Substitute for x in the equation for the # of diagonals:
(x-3) + ((x-3) * (x-2)/2) + (y-3) + ((y-3) * (y-2)/2) = 53
(17 - y - 3) + ((17 - y - 3) * (17 - y - 2)/2) + (y-3) + ((y-3) * (y-2)/2) = 53
(14 - y) + ((14 - y) * (15 - y)/2) + (y-3) + ((y-3) * (y-2)/2) = 53

Step 5: Simplify:
(14 - y) + (y-3) + ((14 - y) * (15 - y)/2) + ((y-3) * (y-2)/2) = 53
(14 - y + y - 3) + ((14 - y) * (15 - y) + (y - 3) * (y - 2))/2 = 53
11 + (210 - 29y + y^2 + y^2 - 5y + 6)/2 = 53
11 + (216 - 34y + 2y^2)/2 = 53
11 + 108 - 17y + y^2 = 53
119 - 53 - 17y + y^2 = 0
66 - 17y + y^2 = 0

Step 6: Use the quadratic formula:
y = (-b +/- (b^2 - 4ac)^(1/2)) / 2a
y = (17 +/- (289 - (4 * 1 * 66))^(1/2)) / 2
y = (17 +/- (289 - 264)^(1/2)) / 2
y = (17 +/- 5) / 2
y = 22 / 2, 12 / 2
y = 11 and 6
Convenient - this gives us both answers!

Check by plugging the answers into our 2 main formulae:
11 + 6 = 17 (Check!)
(x-3) + ((x-3) * (x-2)/2) + (y-3) + ((y-3) * (y-2)/2) = 53
(11-3) + ((11-3) * (11-2)/2) + (6-3) + ((6-3) * (6-2)/2) = 53
(8) + ((8) * (9)/2) + (3) + ((3) * (4)/2) = 53
(8) + (72/2) + (3) + (12/2) = 53
(8) + (36) + (3) + (6) = 53
53 = 53 (Check!)

2007-01-30 00:32:32 · answer #1 · answered by ³√carthagebrujah 6 · 0 0

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