TenMarks teaches you how to apply concepts of similar figures to solve indirect measurement problems.
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Learn about Indirect Measurements In this lesson, let’s learn about indirect measurements using similar triangles. The problem given to us is as follows. A traffic signal, that’s given here, casts a shadow that is 450 cm. long. So, this is the shadow of the traffic signal. Jim, whose height is 200 cm, that’s Jim, standing by the traffic signal and casts a 100 cm long shadow. What is the height of the traffic signal? This is what we need to determine. Let’s learn how we’re going to do this. As we can see, we’ve put down the measurements. This is the length of the shadow for Jim, 100 cm. this is the length of the shadow for the traffic signal, which is 450 cm. Height of Jim is 200, height of the traffic signal, we don’t know. What I’m going to do is simply label these two triangles, A, B, C, and call this D, E, and F. All we did as you can see, I’ve drawn triangles. Since these two triangles are similar, right triangle A, B, C is similar to triangle D, E, F as you can see that, they’re both right angle triangles. If the two triangles are similar because the sun is casting similar shadows on both, then what do we know? Well, if the triangles are similar, then the ratio of corresponding sides is the same. Ratios of the corresponding sides are equal. Which are the corresponding sides, well this side and this side are corresponding. So, the ratio of them which is AB/DE is equal to these two. So, BC/EF=AC/DF. The ratios of all three corresponding sides are equal. Let’s see what we’ve been given. Are we given AB? No, that’s what we need to find out. DE, were given as 200, all units are in centimeters, BC were given as 450, and EF were given as 100. These two were not given but we don’t need that anyway because we’ve got the four values that we need. I see three values and one variable. Let’s call this M. So, if these two ratios or fractions are equal, then their cross products are equal. So, M x 100 should be equal to 450x200, or Mx100=450x200 is 90 followed by three zeros, 90,000. or Mx100=90,000 divided both sides, the left and the right by 100 and equals 90,000/100=900. So, N=900, then this missing side is 900cm because that’s the unit we were using all along. Key things to remember here is all we did was looked at the word problem, converted that to two diagrams. And since these two triangles are similar, we looked at the ratio for the sides. If the ratios aren’t equal, then the product of the ratios, cross product is equal which calculated gives us M or the side equals 900cm. So, the height of this traffic signal is 900 cm.