suppose a triangle has two sides of lengths a and b. if the angle between these sides varies, what is the maximum possible area that the triangle can attain? what can you say about the minimum possible area? could you explain by detail
I did this for you yesterday
http://www.jiskha.com/display.cgi?id=1317265480
I used Calculus to answer the question, clearly the best and easiest way to do the question.
Are you not studying Calculus?
pre-calculus and i didn't understand where the d and cos came from
Ok, draw your base b
Now sketch the other side a , as the height.
You know that the area of the triangle is (1/2)(b)(a)
that is, the multiplication of ab must be as large as possible.
Since the base is fixed, the value of ab would be a maximum when the side 'a' is a vertical line, would you agree with that?
And of course a vertical line suggests an angle of 90° between them
thank you so much for all your help. i understood it now!
To find the maximum possible area of the triangle, we can use the formula for the area of a triangle given two sides and the included angle. The formula is:
Area = (1/2) * a * b * sin(theta)
Here, 'a' and 'b' are the lengths of the two sides of the triangle, and 'theta' is the angle between them. The 'sin' function represents the trigonometric sine of the angle.
To maximize the area, we need to maximize the value of sin(theta). In trigonometry, it is known that the maximum value of sin(theta) is 1, which occurs when the angle theta is 90 degrees or π/2 radians. So, the maximum possible area of the triangle occurs when the angle between the sides is 90 degrees.
At this maximum angle of 90 degrees, the area of the triangle can be calculated as:
Maximum Area = (1/2) * a * b * sin(90 degrees) = (1/2) * a * b * 1 = (1/2) * a * b
So, the maximum possible area of the triangle is equal to half the product of the lengths of the two sides.
Regarding the minimum possible area, the triangle would degenerate into a straight line or a point when the two sides become collinear or coincide with each other. In such cases, the area of the triangle would be zero because it does not exist as a closed shape. Hence, the minimum possible area of the triangle is zero.
To summarize:
- The maximum possible area is equal to half the product of the lengths of the two sides.
- The minimum possible area is zero, which occurs when the two sides are collinear or coincident.