<p>Ok so the question is a diagram so I will just explain it. A triangle is inscribed inside a circle and the longest side of the triangle=the diameter of the circle=17. The 2nd largest side is equal to x+5 while the smallest side is equal to x-2. What does x equal</p>
<p>I think you solve this with inequalities… hold on</p>
<p>is it a specific triangle?</p>
<p>I can email you a picture of the question if you want</p>
<p>No it is not a right triangle or any specific as far as I can tell. I have the picture if the question on my phone and can email to anyone bc I can’t post it here</p>
<p>that would help!</p>
<p>The answer is 7 < x < 12, or x ∈ (8, 9, 10, 11) since the SAT works in integers. The Triangle Inequality holds for these values, and it satisfies the condition that 17 is the longest side.</p>
<p>If a triangle is inscribed in a circle and one of the sides is the diameter, the triangle is a right triangle, with the diameter as the hypotenuse. So just use the Pythagorean Theorem to solve for x.</p>
<p>Yes that’s the right answer but how do you know it’s a right triangle if the largest side is the diameter?</p>
<p>Sojuicy is correct. I remember doing that proof a long time ago in high school.</p>
<p>My answer is correct neglecting the fact that it is in a circle. I misread the question and forgot about that.</p>
<p>It’s very simple. The angle opposite the diameter is an inscribed angle. The measure of an inscribed angle is 1/2 the measure of the intercepted arc. The intercepted arc is a semicircle and thus has measure 180 degrees. So the inscribed angle has measure 90 degrees.</p>
<p>x=10</p>
<p>(x-2)^2 + (x+5)^2 = 17^2
2x^2 + 6x - 260 = 0
x^2 + 3x -130 = 0
(x-10)(x+13) = 0
X has to be positive, so x=10</p>
<p>Also, it’s just a basic triangle fact that if a triangle is inscribed inside a circle and a side is the diameter of the circle, then the triangle is a right triangle and the diameter is the hypotenuse of the triangle.</p>
<p>Couple of things to notice:</p>
<ol>
<li><p>“Number Sense” is really helpful here. As soon as you see the number 17 on the hypotenuse, you should be thinking: 8,15,17 is a triple. Is there an easy way to pick the x value? Oh, yes: make x = 10. Then x-2=8 and x+5 = 15 and it all comes out neat and easy! So even in the calculator age, knowing the triples can be a real time saver. The ones that I try to keep at hand: 3,4,5 and it’s multiples, 5,12,13… 8,15,27 and 7,24,25. </p></li>
<li><p>While it is true that the triangle inscribed in a semi-circle is always a right triangle, I have not seen SAT problems that require this knowledge. They usually stick that right-angle symbol in there for you. So if this was a real SAT question, that’s interesting news…</p></li>
</ol>
<p>That’s 8, 15, 17 — not 8,15,27 sorry for the typo</p>
<p>DumbAndLethal said it well. It should be a right triangle, if a triangle is inscribed in a circle and the one side is a diameter of the circle</p>
<p>Actually, while everything D&L said is true and correct and helpful (I think–I admit, I just skimmed), what pckeller said is very smart (as it often is).</p>
<p>You know that the triangle is a right triangle, for the reason DrSteve stated. You know this is the SAT. And because this is an SAT problem involving a right triangle you should, as pckeller stated, start thinking of Pythagorean triples immediately. </p>
<p>There are 4 primitive Pythagorean triples that you ought to know for the SAT: </p>
<p>3, 4, 5; 5, 12, 13; 8, 15, 17; and 7, 24, 25.</p>
<p>Personally, I never bother stressing the last 2 triples. I’m not sure that these have been useful on more than 1 or 2 SAT questions. And if they were to show up you can always use the Pythagorean Theorem.</p>
<p>The first two come up all the time though, and I have also seen 6,8,10 fairly often (this is just the first triple multiplied by 2).</p>
<p>I personally usually forget those last two and have to use the Pythagorean Theorem to check them. Since I never remember them I don’t ask my students to remember them.</p>
<p>If you want to memorize them, then by all means go right ahead. It’s certainly not a bad thing to do so. But I just want to stress that the first 2 are most important.</p>
<p>If anyone can point to specific SAT questions where the last 2 triples could be used, then that would be helpful.</p>
<p>I agree that 3, 4, 5 and 5, 12, 13 (and their multiples) are more common than 8, 15, 17 and 7, 24, 25.</p>
<p>I don’t find it that hard to remember them, and I don’t have the feeling that they’re taking up space in my brain that I could be using for more useful stuff. (Now, the theme song from The Facts of Life? That is taking up space that I could really be putting to better use.) But if you don’t care to remember the other two, hey, diff’rent strokes…</p>
<p>^Was “Different Strokes” another obscure TV themesong reference? :)</p>
<p>You should replace the Facts of Life theme song with the Different Strokes theme song - a much better song in my opinion. In any case my own brain is cluttered up with both songs. I suppose if you have one you should have both since Facts of Life was a spinoff of Different Strokes.</p>