Which of the fllowing dimensions describes te right triangle with an area of 36cm^2 having the greatest perimeter?
You forgot your choices.
Also -- what do YOU think is the answer?
Of those triangles with integer sides, the right triangle with base equal to 1 and altitude equal to 72 has a perimeter of 145+ square units.
The one with base of 8 ahd altitude of 9 has a perimeter of 29.04 sq. units.
Triangles with sides of 6x12, 4x18 and 2x36 fall inbetween.
To find the right triangle with the greatest perimeter, we need to determine which dimensions maximize the sum of the lengths of its sides.
Let's denote the two legs of the right triangle as 'a' and 'b' and the hypotenuse as 'c'.
We are given that the area of the right triangle is 36 cm^2. The formula for the area of a right triangle is (1/2) * a * b. So, we have:
(1/2) * a * b = 36
Simplifying this equation, we get:
a * b = 72
Now, we want to find the dimensions that maximize the perimeter of the triangle, which is given by a + b + c.
In a right triangle, the Pythagorean theorem states that a^2 + b^2 = c^2.
Solving for c^2, we get:
c^2 = a^2 + b^2
Now, let's substitute the value of 'c^2' from the equation above into the perimeter equation:
Perimeter = a + b + c = a + b + √(a^2 + b^2)
To maximize the perimeter, we can use either calculus or trial and error. Since the given options didn't include specific values, we'll use trial and error.
Let's try out the dimensions given in the options and calculate the perimeters:
Option 1: a = 4 cm, b = 18 cm
Perimeter = 4 + 18 + √(4^2 + 18^2) ≈ 4 + 18 + √(16 + 324) ≈ 4 + 18 + √(340) ≈ 4 + 18 + 18.44 ≈ 40.44
Option 2: a = 9 cm, b = 8 cm
Perimeter = 9 + 8 + √(9^2 + 8^2) ≈ 9 + 8 + √(81 + 64) ≈ 9 + 8 + √(145) ≈ 9 + 8 + 12.04 ≈ 29.04
Option 3: a = 6 cm, b = 12 cm
Perimeter = 6 + 12 + √(6^2 + 12^2) ≈ 6 + 12 + √(36 + 144) ≈ 6 + 12 + √(180) ≈ 6 + 12 + 13.42 ≈ 31.42
Option 4: a = 2 cm, b = 36 cm
Perimeter = 2 + 36 + √(2^2 + 36^2) ≈ 2 + 36 + √(4 + 1296) ≈ 2 + 36 + √(1300) ≈ 2 + 36 + 36.06 ≈ 74.06
From the calculated perimeters, we can see that the right triangle with dimensions a = 2 cm and b = 36 cm has the greatest perimeter of approximately 74.06 cm.
Therefore, the right triangle with dimensions a = 2 cm, b = 36 cm, and c = √(2^2 + 36^2) is the answer.