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The regular octagon has a perimeter of 122.4 cm.
A regular octagon with a radius of 20 centimeters and a perimeter of 122.4 centimeters if shown. Point X is the center point. Lines X Z and X W are radii. Line X Y is an apothem.
Which statements about the octagon are true? Select two options.
The length of segment YZ is 15.3 cm.
The measure of the angle formed by the radius and the apothem is 30°.
The length of segment XY can be found by solving for a in 202 – 7.652 = a2.
The length of segment WZ is 20 cm.
The measure of the central angle, ∠ZXW, is 45°.
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Answers:
- The length of segment XY can be found by solving 20^2 − 7.65^2 = a^2. (True.)
- The measure of the central angle, ∠ZXW, is 45°. (True.)
Work: perimeter 122.4 → side s = 122.4/8 = 15.3 cm, so Y (midpoint of side WZ) gives YZ = 15.3/2 = 7.65 cm. In right triangle X Y Z, XY^2 + 7.65^2 = 20^2, so XY is found from 20^2 − 7.65^2. The central angle = 360/8 = 45°. (The angle between a radius and the apothem is half that, 22.5°; WZ = side = 15.3 cm.)
- The length of segment XY can be found by solving 20^2 − 7.65^2 = a^2. (True.)
- The measure of the central angle, ∠ZXW, is 45°. (True.)
Work: perimeter 122.4 → side s = 122.4/8 = 15.3 cm, so Y (midpoint of side WZ) gives YZ = 15.3/2 = 7.65 cm. In right triangle X Y Z, XY^2 + 7.65^2 = 20^2, so XY is found from 20^2 − 7.65^2. The central angle = 360/8 = 45°. (The angle between a radius and the apothem is half that, 22.5°; WZ = side = 15.3 cm.)
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