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Here is the question:

One side of a triangle is 6 and the other is 7. Which of the following could be the area?

Here are the choices

i)13
ii)21
iii)24

The answers are I only, II only, etc. you know what I'm talking about.

The answer is D which is i and ii.

Why can't it be 24?

By Legendofmax (Legendofmax) on Saturday, October 02, 2004 - 02:37 pm: Edit

Area = .5bh

Right off the bat you could assume that those two lengths were the leg-lengths, yielding an area of 21. This also means 13 can be an answer, since you could have the two legs close together such that their shared angle is really really small, yielding a smaller area overall.

The reason why it can't be 24 is because in terms of the height, it's maximized in right-triangle form, in this case. For instance, if you had 6 as the horizontal base and 7 as the verical leg, rotating that leg in either direction to change the length of the hypotenuse will result in a lowered height value (height defined as the distance from the "top" vertex dropped straight down to the parallel of the bottom leg-base).

By Yujin (Yujin) on Saturday, October 02, 2004 - 05:16 pm: Edit

I'm still kinda confused on your last one.

Why can't the height be say...8?

By Lisasimpson (Lisasimpson) on Sunday, October 03, 2004 - 09:58 am: Edit

because...try to draw out a pic where 2 legs are 7 and 6 and the height is 8 then find the area

By Optimizerdad (Optimizerdad) on Sunday, October 03, 2004 - 02:18 pm: Edit

Here's another explanation that may help.

Since the area of the triangle won't change if you rotate it, we can assume that AB (length=6) is the base. Let BC have a length of 7.
Now picture the triangle with angle ABC slightly less than 180 degrees. BC will be nearly horizontal, and the height (from C down to an extension of the base AB) will be nearly zero. Now slowly decrease the angle ABC towards 90 degrees. As BC rotates, the height gradually increases (as does the area of the triangle), until angle ABC=90 degrees, the height=7, and the area is 0.5(6)(7) = 21. If you now continue to decrease angle ABC from 90 towards 0 degrees, the height (and area) of the triangle will start to decrease, approaching 0 as the angle ABC approaches 0.