TenMarks teaches you how to use intermediate concepts of proportions to solve problems in indirect measurement.
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Learn Intermediate Concepts of Indirect Measurement In this lesson, let’s learn about indirect measurements and how to measure things using properties of similarity. The problem states that a tree, as we can see in the diagram, casts a shadow that is 48 feet long. So we’ve done is drawn the shadow here at 48 feet long and that’s the point PQ. A man, whose height is five feet, and that’s the man here, I’ve tried to draw a diagram that says the man whose height is five feet which is illustrated by TS, casts a shadow that is eight feet long. How tall is the tree? This is what we need to determine. That’s H. So what are we given? We are given two triangles, right. The triangle first one is, TPS. The second triangle is RPQ. All I did was just label them, okay. Since we got two triangles out of it, we just label them triangle TPS and triangle RPQ. Notice that I was careful to make sure that the angles were the same for the small and the big triangle. So P which is the angle is common between the two in the middle. Since these two triangles are similar, the ratio of their corresponding sides is the same, ratio of corresponding sides are equal. What does that mean? Well let’s look at the corresponding sides for both triangles. There is one side that is the big one which is RP. The corresponding side to that in the smaller triangle is TP. So the ratios of this side on the big triangle and side on the little triangle should be the same as these two sides which is PQ on the big triangle and PS on the small triangle, which should be equal to the big height which is PQ or RQ, and the small height which is TS. Now let’s substitute what we already know. What are we given? RP, are we given that? RP is not given. TP is not given. So this is useless. Let’s look at PQ, P and Q, that’s given as 48 feet. PS is given as eight feet equals RQ is what we need to determine so this is the height H and TS is the smaller triangle, and that we're given as five feet. So we’ve got a ratio that’s equivalent to ratios that are equivalent. So the cross products must be equal. So let’s look at the cross products, 48 × 5 = 8 × H. All units are in feet. So 48 × 5 = 8 × H, they’re cross products which means 48 × 5 is 240 = 8 × H. Let’s divide both sides by eight so that I can isolate H on one side. 240 by 8 is 30, 240 ÷ 8 = 30, 8 ÷ 8 = 1 or 8 = 30, since everything is in feet, the height of the tree, since everything is in feet, the height of the tree is 30 feet. That’s what we were looking for. The way we got to it is we just look at this triangle and the big triangle, they were similar. That the triangles are similar than the ratio of their sides are equal. We took these four sides. The ratios give us 48/8 = H/5 which when solved gives us H = 30 or the height brings 30 feet.

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