math question i dont understand

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1

<p>Here's the image:</p>

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<a href="https://satonlinecourse.collegeboard.org/SR/digital_assets/assessment/images/04_MC_VB324908_m02.gif%5B/img%5D"&gt;https://satonlinecourse.collegeboard.org/SR/digital_assets/assessment/images/04_MC_VB324908_m02.gif

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<p>If the five line segments are all congruent, what is the ratio of the length of AC to the length of BD?</p>

<p>A) √2 to 1
B) √3 to 1
C) √2 to 2
D) √3 to 2
E) √3 to √2</p>

<p>I didn't get this question so I looked up the explanation on the SAT Book Owner's website. It said this:</p>

<p>"..the lengths of the side of a 30° - 60° - 90° triangle are in the ratio 1 : √3 : 2.."</p>

<p>How do we know the ratio of the sides based on the angles only? Is there some formula for that or something?</p>

2

<p>You don’t need to know he lengths of the sides. All you need are the ratios, which you can find by using the characteristics of special triangles (30-60-90) and (45-45-90).</p>

3

<p>Oh and the ratios are listed at the beginning of every math section.</p>

4

<p>There are 6 segments there and you say 5 are congruent. You need to tell us which ones.</p>

5

<p>I’m guessing that AB = BC = CD = DA = BD, but I don’t want to assume stuff. If that’s the case, then just use 30-60-90 triangles.</p>

<p>You can determine the ratios of the side lengths given the angles, because all triangles with three given angles are similar to each other. In general, the ratios are determined using the law of sines.</p>

6

<p>^ I agree! The law of sines is an under-appreciated rule. Because of where it falls in the curriculum (and how it is presented), many students don’t realize that it applies to right triangles! And that if you have any trouble with 30-60-90 or 45-45-90 triangles, the law of sines can make all that trouble go away.</p>

<p>(Still, the students who are comfortable with law of sines tend to be the ones who have no trouble with the special right triangles anyway…)</p>

7

<p>They never teach the law of sines applied to 30-60-90 or 45-45-90 triangles because it’s usually easier to just memorize the ratios. But what they forget to teach is that these ratios obviously come from the law of sines. Similarly, the Pythagorean theorem is just a special case of the law of cosines.</p>

8

<p>Which test is this from? I remember doing this</p>

9

<p>@dseed332 It’s from test 9(Blue book) and i saw the same type of question(language modified) in Online Test 5 too.</p>