construct an isoceles triangle with an area of 100. what is the length of the base of the triangle which has the smallest perimeter?
Prepping for a test, explanations please
To construct an isosceles triangle with an area of 100, we need to determine the length of the base that would result in the smallest perimeter.
Let's break down the problem step by step:
Step 1: Recall the formula to calculate the area of a triangle:
Area = (base * height) / 2
Step 2: Since we want to find the length of the base, let's rewrite the formula as:
Base = (2 * Area) / height
Step 3: Now, we need to figure out the height of the triangle. In an isosceles triangle, the height is the perpendicular distance from the base to the top vertex (also called the apex).
Step 4: In an isosceles triangle, the height bisects the base and creates two congruent right triangles. To find the height, we can use the Pythagorean theorem. Let's denote the height as 'h' and the length of the base as 'b' (which we are trying to determine).
Step 5: In the congruent right triangles formed by the height, we know the following:
- The hypotenuse is 'b/2' (half the length of the base).
- The base is 'b'.
- The height is 'h'.
Using the Pythagorean theorem:
(b/2)^2 + h^2 = b^2
Step 6: Simplify and solve for 'h':
b^2/4 + h^2 = b^2
h^2 = b^2 - b^2/4
h^2 = 3b^2/4
h = (√3/2) * b
Step 7: Now, substitute the value of 'h' back into the formula for the base (in terms of 'b') we derived earlier:
Base = (2 * Area) / height
= (2 * 100) / [(√3/2) * b]
= (200 * 2) / (√3 * b)
= 400 / (√3 * b)
≈ 230.94 / b
Step 8: To find the smallest perimeter, we need to minimize the base length 'b'. To achieve this, we set the derivative of the base in terms of 'b' equal to zero and solve for 'b'.
Let's skip the calculus part and find the critical point by observing that (√3 * b) is in the denominator. To minimize the base length, (√3 * b) should be as large as possible.
Step 9: To maximize (√3 * b), we make 'b' as large as possible without exceeding any constraints. In this case, the only constraint is that the base length cannot be negative.
Hence, we set (√3 * b) equal to zero and solve for 'b':
(√3 * b) = 0
b = 0
Since 'b' cannot be negative or zero, we take the next best option, which is b = 0+ (smallest positive number).
Therefore, the length of the base of the triangle with the smallest perimeter is approximately 0.0001 (a very small value).
Please note that in practice, constructing a physical triangle with such tiny measurements may not be feasible, but the mathematically derived value provides a basis for the smallest possible perimeter.