7. which is the side length of the smallest square plate on which a 20-centimeter chopstick can fit along a diagonal without any overhang? round your answer to the nearest tenth of a centimeter.

10 cm
11.5 cm
14.1 cm
28.3 cm

8. the four blades of a helicopter meet at right angles and are all the same length. the distance between the tips of two adjacent blades is 36 feet. how long is each blade? round your answer to the nearest tenth of a foot.

18 ft
20.8 ft
25.5 ft
72 ft

7. 14.1 cm (should be 14.2)

8. 25.5 ft

Step-by-step explanation:

Both of these problems make use of the ratio between the length of the hypotenuse of an isosceles right triangle and the length of its sides.

Consider the right triangle with side lengths 1. Then the hypotenuse is ...

  √(1^2 +1^2) = √(1+1) = √2

The hypotenuse of any isosceles right triangle is √2 times the side length. This means the side length is 1/√2 = (√2)/2 times the length of the hypotenuse. It can be helpful to remember to 3 decimal places √2 ≈ 1.414 and 1/√2 ≈ 0.707.

7. The side length of the plate is (√2)/2 times the chopstick length:

  (√2)/2·20 cm = 10√2 cm ≈ 14.14 cm ≈ 14.1 cm . . . . rounded to 0.1 cm

(Please note that since we have rounded down, the chopstick will actually have some overhang on a plate this size. A plate that will meet requirements will have a side length of 14.2 cm.)

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8. The length of the blade is ...

  (√2)/2·36 ft = 18√2 ft ≈ 25.456 ft ≈ 25.5 ft . . . . rounded to 0.1 ft

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