imhimanshu wrote:Hi,
How do you approach this question
If two sides of triangle are 12 and 8, which of the following could be the area of the triangle
1- 35
2- 48
3- 56
Options are
1 - 1 only
2- 1 and 2 only
3- 1 and 3 only
4- 2 and 3 only
5- 1,2 and 3
OA to follow

To begin, we should recognize that to find the area, we can make any side the base.
So, let's make the side with length 12 be the base.
Now the area = (1/2)(b)(h), so the area will be (1/2)(12)(h) = 6h

Now what are the possible heights of this triangle?

Well, we can move the side with length 8 around to get different heights.

Let's begin with the least height of the triangle. Well, we could move the side with length 8 such that the height is almost zero, which would make the triangle's area almost zero.

What's the greatest height? We will achieve the greatest height by making the side with length 8 perpendicular to the base. This would give the triangle a height of 8 and an area of 48.

So, the range of possible areas is: 0 < area < 48.
This means the answer is [spoiler]B (1 and 2 only)[/spoiler]

In general, if two sides of triangle have lengths x and y, then possible areas of the triangle are: 0 < area < xy/2

Cheers,
Brent