<p>Why is choice C correct in this question?</p>
<p>Sam screenshot pic uploaded in different site: <a href="http://i43.tinypic.com/294gwv6.png%5B/url%5D">http://i43.tinypic.com/294gwv6.png</a></p>
<p>I kinda had the same question. The third side can be anywhere between 4, and 16, inclusive. Why can’t we figure out the are through 7 and the third side for example. Why does it have to be 7 x 10 x 0.5 = 35.</p>
<p>…I don’t understand SirWanksalot explanation of this problem. Can someone explain because I don’t understand either</p>
<p>It has to be 1/2 * 7 * 10 because a triangle’s area is the greatest when it is a right triangle. If you don’t understand, ask me… I’ll explain why this is true…</p>
<p>Explain why its true, please. I figured that was why, but I was trying to prove it and didn’t know where to start.</p>
<p>Visualize it!
When you try to make the hypotenuse larger or smaller, the height is sacrificed.
Here’s a picture so you can see it…</p>
<p><a href=“http://i41.■■■■■■■.com/npi636.jpg[/url]”>http://i41.■■■■■■■.com/npi636.jpg</a></p>
![http://i41.■■■■■■■.com/npi636.jpg[/url]](http://i41.tinypic.com/npi636.jpg%5B/url%5D){kind=link}
<p>I don’t know if you know any linear algebra, but here’s an explanation using methods in that field:</p>
<p>The area of a parallelogram is given by the magnitude of the cross product of vectors specifying two of its sides, so half of that will be the area of a triangle. Let’s specify that we’re starting at the point where the line segment of length 7 and the line segment of length 10 connect. We will call this point A, and the vector along the segment of length 7 we will define to go from A to B, and of length 10 from A to C. Given that, we can say that </p>
<p>Area = .5 | AB x AC |</p>
<p>This equivalent, because of the geometry of the cross product, to the expression</p>
<p>Area = .5 | AB | * | AC | sin (A)</p>
<p>Which in turn is equal to, as we have defined the lengths of the vectors already, the expression</p>
<p>Area = .5 (7) (10) sin(A)
Area = 35 sin(A)</p>
<p>The sine function is bounded by an upper bound of 1 (where A= 90 degrees), so we know that the maximum value of the Area is 35.</p>
<p>You could derive the expression to establish first order conditions, but its completely unnecessary.</p>
<p>Don’t want to use linear algebra?</p>
<p>The area of a triangle is given in some geometry courses to be </p>
<p>Area = .5 a b sin(angle ab)
or
Area = 35 sin(angle ab)</p>
<p>You probably recognize the similarity. Same reasoning applies.</p>
<p>Not good enough? Don’t care about rigor, and just want a common sense approach?</p>
<p>Think of a square. Here, I’ll draw it:
.___
|… …|
|___|</p>
<p>Now, tilt two of the sides, but keep the rest exactly the same:</p>
<hr>
<p>…
.___</p>
<p>Which square will have the most area? To get a hint, let’s take this to an extreme. Tilt the sides all the way. What do we get?</p>
<hr>
<p>What’s the area of this figure? Zero! So we know that tilting it does not add area, and the square has the most area. (Not with rigor, but, eh.) Let’s say that we only concentrate on half the square, i.e., a right triangle. If you tilt the triangle, by the same token, it gets smaller. So, 35 is the max area.</p>
<p>One last type of explanation:</p>
<p>Triangle area is base times height. If you keep the base the same, and tilt one of the sides, which conformation gives the maximum value of the height? Straight up! So we know that in a maximized triangle, base and height are perpendicular, i.e., it is a right triangle. Q.E.D.</p>
<p>Thanks Jimbo!</p>
<p>For whatever reason I was being silly and forgot about A = (1/2)<em>a</em>b*sin(theta)</p>
<p>Makes perfect sense now, though.</p>
<p>Ahhh, thanks so much both JefferyJung and JimboSteve :)</p>
<p>Here is a nice way to visualize that a right triangle gives the maximum area using only your hands. Form an “L” with your two hands - your right hand should be the bottom of the L, and your left hand the side of the L. This simulates a right triangle (right hand=base, left hand=height). Now keep your right hand fixed, and move your left hand back and forth. Notice that the height of the triangle decreases in both directions, thus decreasing the area (since the base is fixed).</p>