One side of a triangle has length 8 and a second side has length 5. Wh

Hello Friends,
Initially, i wasn't sure as to how we can get 3rd side = 2.
Because, |8-5| < 3rd side < |8+5| => 3 < 3rd side < 13. Following image helped...
Now, as skywalker18 correctly mentioned max area of triangle is when it is right triangle.
So, we can't have 3rd side as >= 6 because if so we COULD get 6*8/2 = 24 [NOT POSSIBLE]
so, 3 < 3rd side < 6.

Still based on the diagram area of triange CAN be 5.

Area of a triangle is (1/2) * a*b*sin (angle between a and b)
Since sine has the max positive value 1 and minimum value 0 , hence the area of triangle could be anything between 0 and 20

Your question is not clear. The area of a triangle with sides measuring 5 and 8 could range from 0 to 20 square units (\(0 < area \leq 20\)). The maximum area of 20 square units is achieved when the triangle is a right triangle. Hence, among the given options, the triangle can have an area of either 5 or 20.

Hi manishtank1988
If the minimum length of the third side could be 4 then the minimum area is 1/2*4*5=10 which rules out the possibility of III. Could you elaborate it?
Thanks in advance.

There is no lower limit on the area of any triangle whose only 2 sides are known . Area> 0
So we can also have triangle with area = 5 Skywalker18 can you explain this?

If the minimum length of the third side could be 4 then the minimum area is 1/2*4*5=10 which rules out the possibility of getting III. Could you explain, how can we derive that the lowest area it can go is zero?
Thanks in advance.

The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

Thus, if the lengths of the two sides of a triangle are 5 and 8, then (8 - 5) < third side < (8 + 5). Hence, the length of the third side is between 3 and 13, not inclusive. This, however, does not mean that the minimum length of the third side is 4, as we are not told that the lengths of the sides must be integers.

Next, the area of a triangle with the sides of 4, 5, and 8 is not 1/2*4*5=10, it's \(\frac{3\sqrt {119} }{4}≈8.18...\). The area of a triangle is 1/2*base*height, and since a (4, 5, 8) triangle is not right-angled, its area is not 1/2*4*5 = 10 (the sides with lengths of 4 and 5 are not perpendicular to each other, so they cannot be considered as the height and base).

Finally, when the angle between the sides with the lengths of 5 and 8 approaches 0, the area of the triangle also approaches 0. By adjusting the angle, you can get any area from 0 to 20. (We achieve a maximum area of 20 when the angle between the sides measuring 5 and 8 is 90 degrees. This is because the area, computed using the formula 1/2 * base * height, is maximized when the side measuring 8 is considered the base and the height is maximized. Upon visualizing this, it becomes evident that the height is maximized when it is also equal to the other side of 5.)

Hope it's clear.