<p>Can someone explain how this question works?</p>
<p>The three sides of a triangle have lengths of 6, 10, and w. Which of the following could be the area of the triangle?</p>
<p>I. 12 II. 30 III. 45</p>
<p>Is there only one answer to this problem? If so, then I think the answer is 30. Imagine a right triangle. One leg is 6, the other leg is 10, and the hypotenuse is w. Do a^2 + b^2 = c^2 to get the length of w (the hypotenuse) as square root 136. Once you do the Pythagorean theorm again, you will see that the triangle with the given sides is in fact a righ triangle. To find the area, do base times height. In this case, it doesn’t matter which leg is the base, since you need to multiply the two legs anyway. To find the area, do 6 x 10 x 1/2, which is 30. So the answer is 30. Hope this helped!</p>
<p>It could also be 45. w could range from anything between 5 and 15. If w was 15 and the triangle was a right triangle with legs 15 and 6, then 1/2(6)(15) is 45. I’m not sure about I.</p>
<p>^^ You cannot do that as the height for a 6-10-15 triangle is not 15 (since it isn’t a right triangle). Plus, a leg cannot be longer than the hypotenuse (otherwise it’s not a leg but the hypotenuse). 30 is the only result I can come up with as well. However, I think it is possible to have a triangle of area 12 as well.</p>
<p>I agree with tiger. Darksaber can you be a bit more clear in your explanation?</p>
<p>It is possible to have a triangle of area 12. Taking a base of 10 (base of 6 won’t work if you try it), the height of the triangle would have to be 2.4. Given that another side is 6, use trigonometry to find the length of the remaining side and check if it is between 5 and 15, the inequality I described earlier. I get sqrt((3400-600sqrt(21))/25), which is roughly equal to 5.1, so it satisfies the inequality. Using the same reasoning, you can’t get a triangle of area 45 because the smaller triangles simply wouldn’t work out.</p>
<p>Wharton, darksaber is right because if 15 and 6 were the legs of a right triangle, the hypotenuse could not be 10.</p>
<p>The answer given was 12 and 30, by the way.</p>
<p>Wow, that was a pretty tough question lol at least I got 30 right! :)</p>
<p>If it is a right triangle with sides 6 and 4, then we have an impossible geometrical figure. Who’s to say what’s wrong and what’s right? Can’t we all just play poker on the moon?</p>
<p>Think about it though, if you have a protruding obtuse triangle, 12 is possible. And 30 is just easy (right triangle with sides 6 and 10). 45 is impossible, I believe because the greatest possible area is a right triangle and 6 and 10 being those two sides. I would have to brush up on my geometry to verify that, but if you draw it it’s obvious.</p>
<p>The formula for finding the area of a triangle is A=1/2bh where b is the base length and h is the height.
If 6 and 10 were the base length and height of the triangle, you would do 1/2 times 6 times 10, which would give you 1/2 times 60, which would be 30. There’s one answer.</p>
<p>However, let’s say that 10 is the length of the hypotenuse, which could be possible if w were to equal a number between 6 and 10.
In that case, you would have 1/2 times 6 times w. If w were to equal 4, which it could, then the area would be 12.</p>
<p>The answer 45 cannot work. For the triangle to have an area of 90, you would need to either multiple 10 and 9 or 6 and 15, neither of which work.
A) If you had the numbers 6, 10, and 15 then 15 would be your hypotenuse and would not qualify as either your base length or your height.
B) If you had the numbers 6, 9, and 10 then 10 would be your hypotenuse and would not qualify as either your base length or your height.</p>
<p>I hope that helps! Let me know if you need something elaborated upon.</p>
<p>
I assume you’re talking about a right triangle, as you mention “hypotenuse”. Then, 6 and w would have to be legs. w could not possibly equal 4 because that does not satisfy the Pythagorean theorem. 16 + 36 is NOT equal to 100. I already provided a way to get a solution of 12 above; it’s a lot more difficult that it seems.</p>
<p>Kicks self for not proof-reading and reading replies more carefully.
Thanks for catching my mistake @93tiger16!</p>
<p>The area of the triangle is (1/2)<em>6</em>10*sin A = 30 sin A where A is the angle between the sides of length 6 and 10. Since sin A ranges from 0 to 1, the area of the triangle can range from 0 to 30. So only I and II work.</p>