<p>A kid in my class showed me this problem and i was absolutely stumped.</p>
<p>If a triangle has sides of 7 and 10, what is the greatest possible area of it?</p>
<p>did he tell you the answer?</p>
<p>yea it was a question in some SAT packet the guidance office gave out, the answer was 35, but i have absolutely no idea how that could be an answer.</p>
<p>i remember having this one...except it said it was a right triangle. Ye, i think that has to be a requirement. ANyway, if you just apply the area of a triangle formula for 7 and 10 you get 35. Perhaps the maximized area of a triangle with two given sides is achieved only when it's a right triangle.</p>
<p>I don't completely know why it's right, but I'd say 35.</p>
<p>You know 7 and 10 have to be the side lengths (not the hypotenuse) because you're trying to form the biggest triangle possible, and if 7 and 10 are the sides, the hypotenuse would be even larger than either of those. Right triangles have the most area, so you can guess that the triangle's a right triangle. With right triangles ((side 1)<em>(side2))/2 = area, so (7</em>10)/2 = 35.</p>
<p>Yes, the max area is when it's rght.</p>
<p>The formula for the area of any triangle is Area = (.5)<em>a</em>b<em>sin(theta), where a and b are the lengths of two sides of the triangle, and theta is the angle between the two sides. A triangle with fixed side lengths a and b has maximum area, then, when sin(theta) is at its maximum value: 1. This occurs when theta = 90. Hence, the area of a triangle with two fixed sides always takes its maximum value when the triangle is a right triangle. In this particular instance, .5</em>10*7 = 35, which is why the maximum area is 35.</p>
<p>dang, i never knew that was the formula for any triangle</p>
<p>What happened to "one half base times height" that was drilled into my during tenth grade geometry??? sin theta???</p>
<p>What happens is that b*sin (theta) actually IS the height of the triangle (remember SOH-CAH-TOA?) If you think about it you will realize it is the exact same equation.</p>
<p>@jlime. That is only for right triangles. In Precalc/Calc you learn that the general equation for the area of a triangle with any angle is (1/2)(b)(h*sin(theta)). When theta is 90 degrees (right triangle), sin(theta) becomes 1, and that term drops out, giving you your one-half base times height.</p>