Regular polygon ABCDE is similar to regular polygon TUVWX. Side AB is 5x – 1, and side TU is 4x – 2. These sides are corresponding sides. If the ratio of the perimeter of polygon ABCDE to the perimeter of polygon TUVWX is 4:3, what is the perimeter of polygon ABCDE?
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Solution
Since \(\overline{AB}\) and \(\overline{TU}\) are corresponding sides of similar polygons, the ratio of \(\overline{AB}\) to \(\overline{AB}\) is equal to the ratio of the perimeter of ABCDE to the perimeter of TUVWX; \(\overline{AB}:\overline{TU}=\frac{5x-1}{4x-2}=\frac{4}{3}\), 15x – 3 = 16x – 8, –3 = x – 8, x= 5. Therefore, the length of \(\overline{AB}\) is 5(5) – 1 = 25 – 1 = 24 units. Since ABCDE is a regular polygon with 5 sides, each of the 5 sides measures 24 units. Therefore, the perimeter of ABCDE is (5)(24) = 120 units.
Polygon ABCDEF is similar to polygon GHIJKL. If side AB is 12x + 6x and side GH is 8x + 4x, and these sides are corresponding sides, what is the ratio of the perimeter of polygon GHIJKL to the perimeter of polygon ABCDEF?
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Solution
Since \(\overline{AB}\) and \(\overline{AB}\) are corresponding sides of similar polygons, the ratio of \(\overline{AB}\) to \(\overline{AB}\) is equal to the ratio of the perimeter of GHIJKL to the \(\frac{8x+4x}{12x+6x}=\frac{12x}{18x}=\frac{2}{3}\)= 2:3.
If the ratio of the perimeter of octagon ABCDEFGH to the perimeter of octagon STUVWXYZ is 1:1, which of the following must be true?
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Solution
The number of sides of a polygon determines the sum of the interior angles of the polygon. The sum of the interior angles of any octagon is 180(8 – 2) = 180(6) = 1,080°. These two octagons are not necessarily similar or congruent, although they could be. The ratio of \(\overline{AB}\) to \(\overline{ST}\) could be 1:1, but there could also be no ratio between the sides of ABCDEFGH and STUVWXYZ. No one side of ABCDEFGH must equal one side of STUVWXYZ.
Quadrilaterals ABCD and EFGH are similar. If the perimeter of quadrilateral ABCD is equal to 4y2, what is the perimeter of quadrilateral EFGH?
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Solution
Although it is known that ABCD and EFGH are similar, the ratio of their corresponding sides is not provided, nor is the ratio of their perimeters. Therefore, the perimeter of EFGH cannot be determined.
Pentagons ABCDE and FGHIJ are similar. The ratio of each side of pentagon ABCDE to its corresponding side of pentagon FGHIJ is 4:1. If \(\overline{AB}\) and \(\overline{FG}\) are corresponding sides, and the length of \(\overline{AB}\) is 4x + 4, what is the length of \(\overline{FG}\)?
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Solution
The ratio of \(\overline{AB}\) to \(\overline{FG}\) is 4:1. Therefore,4⁄1 = \(\frac{4x+4}{FG}\), 4(\(\overline{FG}\)) = 4x + 4, FG= x + 1.
The sum of the interior angles of a polygon is 9x2. If x is 3 greater than the number of sides of the polygon, how many sides does the polygon have?
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Solution
The sum of the interior angles of a polygon is equal to 180(s – 2), where s is the number of sides of the polygon. Since the number of sides is three less than x and the sum of the interior angles is 9x2, 180(x – 3 – 2) = 9x2, 180x – 900 = 9x2, 9x2 – 180x + 900 = 0, x2 – 20x + 100 = 0, (x – 10)(x – 10) = 0, x = 10. Therefore, the number of sides of the polygon is (10) – 3 = 7 sides.
If the sum of the interior angles of a polygon is equal to the sum of the exterior angles, which of the following statements must be true?
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Solution
The sum of the exterior angles of any polygon is 360°. The sum of the interior angles of a polygon is equal to 180(s – 2), where s is the number of sides of the polygon. If these sums are equal, then 180(s – 2) = 360, 180s – 360 = 360, 180s = 720, and s = 4. The polygon has 4 sides.
Andrea draws a polygon with x number of sides. The sum of the interior angles of her polygon is 60 times its number of sides. How many sides does Andrea’s polygon have?
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Solution
The sum of the interior angles of a polygon is equal to 180(s – 2), where s is the number of sides of the polygon. If x is the number of sides of the polygon, and the sum of the interior angles is 60 times that number, then 60x = 180(x – 2), 60x = 180x – 360, 120x = 360, x = 3. Andrea’s polygon has three sides
The sum of the interior angles of a polygon is equal to three times the sum of its exterior angles.How many sides does the polygon have?
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Solution
The sum of the exterior angles of any polygon is 360°. Therefore, the sum of the interior angles of this polygon is (360)(3) = 1,080. The sum of the interior angles of a polygon is equal to 180(s – 2), where s is the number of sides of the polygon; 1,080 = 180(s – 2), 1,080 = 180s – 360, 1,440 = 180s, 144 = 18s, s = 8. The polygon has 8 sides.
Heather draws two regular pentagons. Which of the following is NOT always true?
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Solution
Regular pentagons are equilateral; every side is equal in length. Therefore, every angle is equal in size, and every regular pentagon is similar to every other regular pentagon. However, regular pentagons are not necessarily congruent. One regular pentagon could be ten times the size of another. Heather’s regular pentagons may not be congruent.