<p>In a ABC triangle, two side AB and BC are equal. Side AC has a length of 5, and the perimeter of ABC triangle is 13. If the measure of angle ABC is x degrees, which of the following must be true about x?</p>
<p>A. x<60
B. 60<x<90 c.="" x="90" d.="">90
E. It cannot be determined from the information given.</x<90></p>
<p>The right answer is B, but I put E. How can we determine the angle if an isosceles triangle can have many different angle options, such as an obtuse isosceles or acute isosceles.</p>
<p>So, there’s obviously a certain relation between the angles. Consider this. Consider that AB = 3. What lengths do we have now? 3, 4, and 5, right? Therefore, the measure of angle ABC is 90 degrees. But the length of AB is NOT 3. Infact, the length of AB is 4. Therefore, angle ABC is less than 90 degrees. But also, AB, and BC are equal, remember? That means two angles are equal, and there’s a third angle here in the triangle which is not equal to 90. Can one of the angles of the triangle = 60? No. Why? Because the three sides aren’t equal. If one angle equals 60, then the other 2 have to equal 60 each as well, which is impossible, since the sides of this triangle aren’t equal.</p>
<p>Therefore, angle ABC or angle X is greater than 60, but less than 90. Choice B is correct.</p>
<p>Hinge theorem: Bigger angles make larger opposite sides. (makes intuitive sense, can be proven with Law of Sines)</p>
<p>We know it’s less than 90 degrees because if it was 90 degrees, side AC would be 4 sqrt(2) by the pythagorean theorem. That’s a lot more than 5, so angle ABC doesn’t open up that wide. If it was 60 degrees, all the sides would be equal, so side AC would be 13/3 which is less than it really is; thus angle ABC opens up wider than 60 degrees.</p>
<p>Don’t say it can’t be determined! Since you know the side lengths, the angles are uniquely determined (SSS congruence). In fact, if you have enough time, you can find the exact angle using the Law of Cosines. (or, if you want to be underhanded, you can draw it yourself and look at how big the angle is!)</p>
<p>since the perimeter is 13, one side=5 and the other 2 sides are equal, we know that AB=BC=4
If you draw the height from point B, it also cuts angle ABC in 2.
therefore SIN(x/2)=(5/2)/4=5/8 -> x/2=38.7 DEG
x=77.36 DEG. The Answer is B</p>
<p>@Jeffery: This isn’t entirerly true. On a Level 1, 2, or 3 question, the answer is sometimes “it cannot be determined.” On a Level 4 or 5 problem it is very unlikely that this will be the answer (in fact, I’ve never seen it happen).</p>
<p>I haven’t really gone through many real SAT as you probably have, so I’m glad to concede my statement earlier. However, I personally have never seen a real QAS that had “it cannot be determined” as the answer on any of the problems. I do recall seeing this on PR and Barron’s SAT books.</p>
<p>I would say that on Level 1, 2 and 3 problems this will come out as an answer roughly 1/5 of the time (consistent with the answer choices being uniformly distributed). Therefore if you’ve only looked at a few actual tests it’s unlikely you will see any of these cases.</p>
<p>Again, I have NEVER seen it come out as the answer on a Level 4 or 5 question (this doesn’t mean it can’t happen, just that it is statistically VERY unlikely). That’s why I always tell my students to (quasi-)eliminate this choice if it comes up on the last third of the SAT.</p>
<p>I took the Nov SAT and there was one answer that was “it cannot be determined”. I believe it was number 9 for a section with 19 questions. Is this a level 3 question? Anyway, I take back what I said in the earlier post…</p>