the hypotenuse of the right triangle is 20 cm. Determine the perimeter that will make its area minimum?
half of a square will max area.
sorry.
Naturally, an area of zero will be a minimum. So, pick sides of 0 and 20.
That's technically not a triangle, but you van pick any tiny value h and make the sides
h and √(400-h^2)
and the area will be as close to zero as you like.
To determine the perimeter that will make the area of the right triangle minimum when the hypotenuse is 20 cm, we need to find the minimum value of the area.
Let's assume the two legs of the triangle are a and b (where a is the shorter leg and b is the longer leg).
By applying the Pythagorean theorem, we have the equation:
a^2 + b^2 = (20 cm)^2
Now, we need to express the area of the triangle in terms of a and b. The area of a triangle is given by:
Area = (1/2) * a * b
Let's substitute the value of b from the Pythagorean theorem equation into the area equation:
Area = (1/2) * a * (sqrt((20 cm)^2 - a^2))
To find the minimum area, we can take the derivative of the area equation with respect to a and set it equal to zero:
d(Area)/da = 0
Differentiating the equation, we get:
(1/2) * (sqrt((20 cm)^2 - a^2)) - (1/2) * (a^2 / sqrt((20 cm)^2 - a^2)) = 0
Now, we can solve this equation to find the value of a that will minimize the area. However, this involves complex calculations, and it is more efficient to use calculus optimization techniques.
To summarize, to determine the perimeter that will make the area of the right triangle minimum when the hypotenuse is 20 cm, we need to find the derivative of the area equation with respect to a and solve for the value of a that minimizes the area.
To determine the perimeter that will make the area of the right triangle minimum, we need to find the other two sides of the triangle first. Given that the hypotenuse is 20 cm, let's label the other two sides as "a" and "b".
The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the hypotenuse.
So, we have the equation:
a^2 + b^2 = 20^2
Now, the area of the right triangle is given by the formula:
Area = 0.5 * base * height
In a right triangle, if we consider the two shorter sides (legs) as the base and height, then the area formula becomes:
Area = 0.5 * a * b
We want to minimize the area, which means we want to minimize the product of "a" and "b".
To find the perimeter, we need to add all three sides of the triangle:
Perimeter = a + b + 20
Now, to minimize the area, we need to find the minimum product of "a" and "b". This happens when "a" and "b" are equal, giving a square shape to the triangle.
So, let's set "a" equal to "b":
a = b
Now, substitute "b" in the equation for the area:
Area = 0.5 * a * a
Area = 0.5 * a^2
To find the minimum area, we can take the derivative of the area function with respect to "a" and set it equal to zero:
d(Area)/d(a) = 0
0.5 * 2a = 0
a = 0
Since "a" cannot be zero, there is no minimum area.
Therefore, there is no specific perimeter that will make the area of the right triangle minimum.