<p>What is the greatest possible area of a triangle with one side of length and another side of length 7 and another side of length 10?
(A)17<br>
(B)34
(C)35
(D)70
(E)140</p>
<p>I picked D because if you were to make a trianlge with sides 10, 14, and 7, you'd get 70 as its area. </p>
<p>Why does the anwer have to be C?</p>
<p>You’re very close. You need to remember that area of a triangle is base<em>height</em>1/2. I think you forgot to multiply by .5.</p>
<p>I did 5*14=70</p>
<p>Right, but that would make your base 10 and your height 14. This would be a right triangle whose hypotenuse is the shortest leg–not possible. What you need to realize about these problems is that the 2 lengths given in the problem are the base and the height of the right triangle.</p>
<p>So, Area=base<em>height</em>1/2=.5<em>7</em>10=35</p>
<p>How do you know if they’re talking about a right triangle? It’s possible to form acute and obtuse triangle with those two sides.</p>
<p>True, but though the base will increase, the height will decrease. It’s just important to remember that the right triangle will yield the largest area.</p>
<p>It has to be a right triangle because the question asks for the “greatest possible area of a triangle.” A right triangle being the largest area is just one of those known shortcuts one can rely on. Something of the same nature might appear on tests such as the GMAT or GRE, but with squares. It is, however, NOT needed to know this to solve the problem. If you do, that is great, if not, it’s no biggie!</p>
<p>In the simplest terms, just think that a right triangle does maximize the possible values of the base and the height. You can play around with some sketches of possible triangles and it will make sense. Remember the basic rules for the sizes of the sides. </p>
<p>Another alternative is to check your basic trig for the area of a triangle and the maximum values of sin, but that goes beyond the SAT.</p>