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Ratio plays an important role in measurement and can be used to make predictions. If the ratio of inches to centimeters is 1 to 2.54 (1:2.54), then we can assume that a length of 12 in. is approximately 30 cm (30.48).
Not all ratios in nature are constant, though. According to mathematician Ernest Zebrowski Jr., "Most ratios, in fact, are not constant. If, for instance, it took 24 rowers to row a galley at 15 mi/h, this does not mean that 48 rowers would get the boat up to 30 mi/h and that with 144 rowers the boat would hit 90 mi/h. (In fact, this line of reasoning would suggest that the ancients could have broken the sound barrier just by getting together enough rowers.) .... Although it's a simple matter for an accountant or mathematician to assert that a particular ratio is constant, the laws of nature are the final arbiter. Clearly, before making predictions on the basis of an assumed constant ratio, we need to get someone to check out the reality of the situation."
While Zebrowski states that many ratios are not constant, there are some constant ratios found in measurement situations. One constant ratio that we use regularly is  . We will explore this ratio further in Session 7, which focuses on circles.
Another common measurement situation involves right triangles. We will now look more closely at right triangles, beginning with a number of right triangles of different sizes.
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Print the shapes from the PDF file (be sure to print this document full scale). With a centimeter ruler, measure the hypotenuses of these triangles. We will explore whether there is a constant of proportionality.
Complete the chart. (Notice that these are all isosceles right triangles.):
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Side Lengths (S) in cm |
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Hypotenuse Length (H) in cm. |
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Ratio S:S |
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Ratio H:S |
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1 |
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1.4 |
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1:1 |
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1.4:1 |
2 |
2.8 |
2:2 |
2.8:2 |
3 |
4.2 |
3:3 |
4.2:3 |
4 |
5.7 |
4:4 |
5.7:4 |
5 |
7.1 |
5:5 |
7.1:5 |
6 |
8.5 |
6:6 |
8.5:6 |
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hide answers |
Answers may vary due to measurement. Here, answers are given to the nearest tenth of centimeter.
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