Video: Using Right Triangle Trigonometry to Find Lengths in Word Problems

A man who is 1.7 meters tall is standing in front of a 4.3 meters high lamp post. When the lamp post is turned on the man’s shadow is 2.2 meters long. Find the distance between the man and the base of the lamp post giving the answer to two decimal places.

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Video Transcript

A man who is 1.7 metres tall is standing in front of a 4.3 metres high lamp post. When the lamp is turned on, the man’s shadow is 2.2 metres long. Find the distance between the man and the base of the lamp post, giving the answer to two decimal places.

We know that the lamp post is 4.3 metres tall and that the man is 1.7 metres tall. When the light is turned on, the man cast a shadow on the ground that is 2.2 metres long. We want to know how far the man is from the lamp post. Let’s call that 𝑥.

There are a few ways for us to solve this. But both of them involve us thinking about this in terms of right-angled triangles. Our image is obviously not drawn perfectly to scale. But we should see that we have a large right triangle that has the height of the lamp post 4.3. And then, it has a shadow of 2.2 metres and 𝑥 metres as its other side length. Inside the larger triangle, there’s a smaller triangle who has one side length of 1.7 metres and another side length of 2.2 metres. These triangles share an angle measure. We can call it angle 𝜃.

Using some trig ratios, we could solve for the measure of angle 𝜃. We know the ratios of sine, cosine, and tangent. Sine is the opposite angle over the hypotenuse. The cosine is the adjacent angle over the hypotenuse. And the tangent ratio is the opposite over the adjacent. Looking at our smaller triangle, we know the opposite side length and the adjacent side length, which means we could set up a tangent ratio. We could say that the tangent of angle 𝜃 is equal to 1.7 over 2.2.

In order to solve for 𝜃, we need to take the tangent inverse of both sides of the equation, sometimes called the arctan. Tangent inverse of tangent of 𝜃 just equals 𝜃. And if we plug in the tangent inverse of 1.7 over 2.2 into our calculators, we’ll get 37.69424 continuing and we don’t want to round yet. We can use this angle measure to solve for our missing variable in the larger triangle. This will be another tangent ratio. This time, we can say that tangent of 37.69424 continuing equals the opposite side length of 4.3 metres. The adjacent side length of the larger triangle is equal to this distance. And that means for our denominator, we need to say it’s going to be 𝑥 plus 2.2. It’s 2.2 metres plus the missing distance from the man to the lamp post.

We’ll plug in tangent of 37.69424 continuing, which gives us 0.772 repeating. Again, we won’t do any rounding in this step. We’ll just bring down the right-hand side 4.3 over 𝑥 plus 2.2. At this point, we want to get 𝑥 out of the denominator. So we multiply both sides of the equation by 𝑥 plus 2.2. And then, we’ll have 𝑥 plus 2.2 times 0.772 repeating is equal to 4.3. From there, we can divide both sides of the equation by 0.772 repeating. So we have 𝑥 plus 2.2 equals 5.564705 continuing. And then, we subtract 2.2 from both sides, which gives us 𝑥 equals 3.364705 continuing.

To round this value to two decimal places, we’ll consider the third decimal place. There’s a four in the third decimal place. So we’ll round this value down to 𝑥 equals 3.36. Remember, we’re working in metres. And that 𝑥-value is the distance from the man to the lamp post. So we can say that the man is 3.36 metres from the base of the lamp post. But remember at the beginning of the video, I said there was more than one way to solve this problem. So let’s go back.

We can actually solve this problem without using the sine, cosine, or tangent ratios at all. Here’s what we need to know. Number 1) In two triangles, if two pairs of angles are congruent, then the triangles are similar. And we call this angle-angle similarity. These two triangles share the angle 𝜃 and they both have a right angle. And so, we can say that these two triangles are similar. We also know that in similar triangles, corresponding sides are always in the same ratio.

Based on this information, we can set up a ratio. In our smaller triangle. We have 1.7 over 2.2. The corresponding side in the larger triangle for 1.7 is 4.3 and the corresponding side to 2.2 in the larger triangle is 𝑥 plus 2.2. First step to solving this we will cross multiply. And we’ll have 𝑥 plus 2.2 times 1.7 is equal to 2.2 times 4.3. 2.2 times 4.3 is 9.46. Bring down the left-hand side and then divide both sides of the equation by 1.7. On the left, we’ll have 𝑥 plus two. And on the right, 9.46 divided by 1.7 equals 5.564705 continuing. When we subtract 2.2 from both sides, we get 𝑥 is equal to 3.364705 continuing. And that confirms our original findings that the man is standing 3.36 metres from the base of the lamp post.

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