kartheekchoudhary wrote:One side of a triangle has length 8 and a second side has length 5. Which of the following could be the area of the triangle?
I 24
II 20
III 5
D
The biggest triangle possible will be formed if the known sides (8 and 5) are the base and height of a right triangle. To illustrate, imagine that the triangle lays flat on its side of 8. The largest triangle possible will be formed with the longest height available. This is achieved if the side of 5 stands vertically. If 5 isn't straight up but slanted, then the height will in fact be less than 5. Thus the greatest triangle has base=8 height=5, area=20.
You may wonder why the 3rd unknown side cannot be used as the height (or base). The answer is simple; in right triangles, the hypotenuse is the longest side. If we make side of 8 be the hypotenuse, then the 3rd unknown side will be smaller than 8. 5 cannot be the hypotenuse because it's shorter than 8. To maximize area, we need the greatest hypotenuse possible. As we built the triangle in the previous paragraph, the 3rd unknown side is the hypotenuse, so it's longer than 8 and 5.
krazy800 wrote:could you please explain how option III or 5 could be true?
There is no lower bound on the area of any triangle when you only know two sides (actually area > 0). This is because the area will be decided by the angle between the two known sides. Above I explained why a 90 degree angle creates the maximum area. As this angle gets smaller (60 degrees) and smaller (15 degrees), you can visualize the area shrinking until it approaches 0 when the angle approaches 0 degrees. So the area could be 5, 3, 1, or 0.0003.
If you have trouble with these types of Qs, you can use the Drill Engine to generate timed drills with similar questions; set topic='Geometry' and difficulty='600-700 & 700+'
Hope that made sense,
-Patrick
