Section 6 #9

Quantity A : The length of the line segment AC Quantity B : 3 Quantity A is greater., Quantity B is greater., The two quantities are equal., The relationship cannot be determined from the information given.

2 Explanations

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Jacob Elder

So you described BD as x at one point and also as x root 3.

So I was confused about the ratios given that BD is part of both triangles. Since 45 45 90 triangles have x, x, x root 2 ratio, BD should presumably be x, right? But then 30 60 90 triangles have an x, 2x, x root 3 ratio too. So how do you determine whether BD is either x or x root 3, because the two ratios suggest conflicting values for that side, it seems. Evidently from the video, it seems you determined that BD is x root 3 rather than x, but I was wondering why you determined that instead of x?

Jan 28, 2016 • Comment

Cydney Seigerman, Magoosh Tutor

Hi Jacob :)

I can see how the explanation video can be a little confusing. First, let's review the properties of the 30-60-90 triangle and then look at the practice problem again :)

For the 30-60-90 triangle, we have the following ratios:

angles: 30 : 60 : 90
side lengths: x : x*sqrt(3) : 2x

where the longest side (2x) is across from the 90 angle.

With that in mind, let's look at the practice problem. BC is the hypotenuse of BCD and therefore will have a length of 2x according to the ratios above:

2x = 2
x = 1

This means that the DC has a length of 1 and BD a length of sqrt(3).

Now, let's consider the other right triangle, the 45-45-90 triangle, and to help us keep things clear, instead of using x to discuss ratios, let's use a different variable, w:

AD = BD = w

From our evaluation of BDC, we know that BD has a length of sqrt(3). This means that w = sqrt(3) and that AD = sqrt(3).

Since we know AD and DC, we can find the length of AC:

AD = sqrt(3)
DC = 1
AC = AD + DC = 1 + sqrt(3)

Knowing that the length of AC is 1 + sqrt(3), we can definitively say that Quantity B is greater, since 3 > 1+sqrt(3).

I hope this helps! :)

Feb 17, 2016 • Reply

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Chris Lele