Solve each problem.
Length of a Road A camera is located on a s...

Question

Solve each problem. 
 Length of a Road A camera is located on a satellite with its lens positioned at C in the figure. Length PC represents the distance from the lens to the film PQ, and BA represents a straight road on the ground. Use the measurements given in the figure to find the length of the road. (Data from Kastner, B., Space Mathematics, NASA.)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to utilize the provided measurements in the following diagram to solve the following problem. We're given a diagram that essentially has two triangles that are connected at point P. We're told the diagram is not drawn to scale. Then we have a description of what the diagram is attempting to show. We're told a telescope is positioned on top of a mountain with its lens located at point P. OK. So triangle PUV is a mountain, we can add some snow and maybe a skier going down the side of the mountain. And we're told that there's a telescope on top of the mountain at point P. So essentially that triangle A BP is the telescope P A represents the distance from the lens to the photographic plate. OK. So the distance from the lens to the photographic plate is A B or excuse me A P or P A which is 390 centimeters and A B is 7.8 centimeters. We can label that that's a photographic plate if we can spell photo graphic plate and up here's the lens while UV represents the length of a river flowing a straight path in the valley. All right, at the base of our mountain. UV, that's a river. Maybe we've got a fish in it. The only other side length that we're given is pu which is the distance from the peak of the mountain to you where the river is. And that's 25 kilometers. Determine the length of the river. OK. That's what we're doing here. We're finding UV that fourth side. Our answer choices are answer choice, a five kilometers, answer choice B 125 kilometers. Answer choice, C 1250 m and answer choice D 500 m. All right. So taking a look at our diagram, when we have two triangles, we know three sides and we want to figure out 1/ side, we can do so if the two triangles are similar triangles. So do we have similar triangles here? Well, we recall from previous lessons that if we have two angles from one triangle similar to two angles from another triangle that they are similar triangles from the A, a postulate for the angle angle postulate. So looking at our two triangles, they share that point P which means that UV or excuse me, U PV and A PB are vertical angles which by definition are congruent. So those two are definitely similar. Now, do we have two other ones that are similar? Well, if we make an assumption that the person standing on top of the mountain is holding their photographic plate parallel with the river and it looks like they are, then we would have angle a similar to angle U, they even look like they're both right angles. So making that assumption, then we would have two similar triangles and then we can proceed to set up ratios to figure out that fourth side. When we have a fraction equal to a fraction for our ratios, each triangle gets its own fraction. So we can say that the first triangle or excuse me, the first fraction can be for the telescope and the second fraction can be for the mountain or in other words, the first fraction is for triangle A PB. And the second fraction is for triangle UVP. And then the numerators from each of these are going to be similar sides and the denominators from each of these are going to be the other two similar sides. So for the numerators, let's get the parts of line A U for the telescope triangle, the part of the line A U is A P and the length of A P is 390 centimeters. And for the mountain triangle, the part of the line A U is pu, the length of PU is 25 kilometers. All right. Now, for our denominators, those are going to be those supposedly parallel lines, the photographic plate and the river. So for the telescope triangle, that's going to be the line A B which we said was 7.8 centimeters. And for the mountain triangle, that's going to be the line UV, which is what we're trying to find. We do not know that now that we have the ratio set up, all we have to do is cross multiply to solve for UV. So a after all that, we just need to do a little bit of algebra. So we have 7.8 centimeters multiplied by 25 kilometers and that's equal to 390 centimeters multiplied by UV. To get UV by itself. We're going to divide both sides by centimeters. And we see that those centimeter units will cancel out in the numerator 7.8 multiplied by equals 195. So we have 195 kilometers divided by equal to UV on the right and 195 divided by 390 equals 0.5 kilometers. All right. So the length of that river is 0.5 kilometers. Now, moving this to where our answer choices are, wouldn't it be nice if we're done? But none of the answer choices match 0. kilometers. Our kilometers answers are five and 125. But we have two answers in meters. So does this equal an answer in meters? We can multiply this by a ratio and do some unit conversion here. So how many meters in a kilometer? It's 1000 m in a kilometer. So we put one multiplying our 0.5 kilometers by 1000 m divided by one kilometer, our kilometer units cancel out. And if we take 0.5 multiplied by 1000 that gets us 500 it'll be 500 m which does match an answer choice that matches with answer choice. D well done. We'll catch you on the next one.

Key Concepts

Trigonometric Ratios

Right Triangle Properties

Angle of Elevation and Depression

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