Question Video: Dividing a Triangle into Right Triangles in Order to Calculate an Unknown Length | Nagwa Question Video: Dividing a Triangle into Right Triangles in Order to Calculate an Unknown Length | Nagwa

Question Video: Dividing a Triangle into Right Triangles in Order to Calculate an Unknown Length Mathematics • Third Year of Preparatory School

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In the figure ๐ด๐ถ = 3.5. What is ๐ด๐ต? Give your answer to two decimal places.

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Video Transcript

In the figure, the length of ๐ด๐ถ is 3.5. What is the length of ๐ด๐ต? Give your answer to two decimal places.

In this question, weโ€™re given a figure of triangle ๐ด๐ต๐ถ, and weโ€™re told that the length of side ๐ด๐ถ is 3.5. We can add this to our diagram. We need to determine the length of the side ๐ด๐ต. We need to give our answer to two decimal places. Thereโ€™s a few different ways we could go about determining the length of ๐ด๐ต.

To do this, we note that in our diagram weโ€™re given the point ๐ท which makes two right triangles. We have right triangle ๐ด๐ท๐ต and right triangle ๐ด๐ท๐ถ. And if we add the right angle for triangle ๐ด๐ท๐ถ, we note that we have one known side length for this triangle and we know one of the nonright angles of the triangle. This means we can determine the side lengths of this triangle by using right-triangle trigonometry. And in particular, we can use this to find the length of ๐ด๐ท. And then, weโ€™ll be able to use right-triangle trigonometry on our other right triangle ๐ด๐ท๐ถ to determine the length of ๐ด๐ต.

The first thing weโ€™re going to need to do is label the sides of our right triangle ๐ด๐ท๐ถ. Letโ€™s start with the hypotenuse. Itโ€™s the longest side in the right triangle. Itโ€™s the one opposite the right angle. In this case, thatโ€™s side ๐ด๐ถ. Next, we should label the side we want to calculate; thatโ€™s ๐ด๐ท. We can see in our diagram itโ€™s opposite the angle of 41 degrees. So we can label this as the opposite side. Finally, although itโ€™s not necessary, we can also see that side ๐ท๐ถ is adjacent to our angle of 41 degrees. So we can label this as the adjacent side.

Now that weโ€™ve labeled the sides of this right triangle, weโ€™re ready to apply right-triangle trigonometry to determine the length of side ๐ด๐ท. And we can recall the following acronym: SOH CAH TOA. This will help us determine which trigonometric ratio we need to use. In our diagram, we want to determine the opposite side, and we know the length of the hypotenuse. So we need to use the trignometric ratio, which relates the opposite side to the hypotenuse. And this is the sine function. We can recall if ๐œƒ is the angle in a right triangle, then the sin of ๐œƒ will be equal to the length of the opposite side to angle ๐œƒ divided by the length of the hypotenuse. Now, weโ€™d substitute the values from our right triangle ๐ด๐ท๐ถ into this equation. We get sin of 41 degrees is equal to the length of ๐ด๐ท divided by 3.5.

Now, we can solve for ๐ด๐ท by multiplying both sides of our equation through by 3.5. We get that the length of side ๐ด๐ท is 3.5 times the sin of 41 degrees. And now we could use our calculator to evaluate this expression. However, remember, weโ€™re trying to find the length of side ๐ด๐ต. And weโ€™re going to need to use the length of side ๐ด๐ท to do this, so itโ€™s easier to use the exact value. So weโ€™ll leave this expression as it is.

Letโ€™s now sketch triangle ๐ด๐ต๐ท. We have the triangle ๐ด๐ท๐ต is a right triangle. We know one of the nonright angles of this triangle and we also know one of the side lengths. So we can determine the length of this triangle by using trigonometry. Once again, weโ€™ll start by labeling the sides of this triangle. Weโ€™ll start with the hypotenuse, the longest length of the triangle, the one opposite the right angle. In this case, thatโ€™s ๐ด๐ต. Next, we see that the side ๐ด๐ท is the one opposite our angle of 63 degrees. So we can label this as the opposite side. And finally, although itโ€™s not necessary, we can see that the side ๐ต๐ท is the one adjacent to the angle of 63 degrees. So we can label this as the adjacent side.

We now need to determine which trigometric ratio we need to use. And to do this, we will once again use our acronym of SOH CAH TOA. We have a very similar story. However, it is slightly different in this case. In triangle ๐ด๐ต๐ท, we know the length of the opposite side to our angle. However, we want to determine the length of the hypotenuse. So weโ€™ll once again be using the sine ratio. Now, all we need to do is substitute the values from triangle ๐ด๐ต๐ท into the sine ratio. We get sin of 63 degrees will be equal to 3.5 sin of 41 degrees divided by the length of ๐ด๐ต.

Now, all thatโ€™s left to do is rearrange this equation for ๐ด๐ต. Weโ€™ll multiply both sides of our equation through by ๐ด๐ต and divide both sides of the equation through by sin of 63 degrees. This gives us that the length of ๐ด๐ต is equal to 3.5 times the sin of 41 degrees divided by the sin of 63 degrees. And now, we can evaluate this expression by using our calculator, where we need to make sure itโ€™s set to degrees mode. If we do, we get 2.577 and this expansion continues. Well, itโ€™s worth pointing out we could also give this the units of length units since we know this represents the length.

Finally, the question wants us to write our answer to two decimal places. To do this, we look at the third decimal digit of our number, which is seven. Since this is greater than or equal to five, we need to round this value up. And in doing so, we get our final answer. Given in the figure that ๐ด๐ถ is 3.5, we were able to conclude the length of ๐ด๐ต is 2.58 to two decimal places.

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