Find the maximum area of a triangle with a fixed perimeter of 30 cm. (2022)

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Question: Find the maximum area of a triangle with a fixed perimeter of 30 cm. (2022)

Options:

Correct Answer: 75 cm²

Exam Year: 2022

Solution:

For maximum area, the triangle should be equilateral. Area = (sqrt(3)/4) * (10)^2 = 75 cm².

Find the maximum area of a triangle with a fixed perimeter of 30 cm. (2022)

Find the maximum area of a triangle with a fixed perimeter of 30 cm. (2022)

Step 1: Understand that we want to find the maximum area of a triangle with a fixed perimeter of 30 cm.
Step 2: Recall that for a triangle with a fixed perimeter, the maximum area is achieved when the triangle is equilateral.
Step 3: Since the perimeter is 30 cm, divide this by 3 to find the length of each side of the equilateral triangle: 30 cm / 3 = 10 cm.
Step 4: Use the formula for the area of an equilateral triangle: Area = (sqrt(3)/4) * (side length)^2.
Step 5: Substitute the side length (10 cm) into the area formula: Area = (sqrt(3)/4) * (10 cm)^2.
Step 6: Calculate (10 cm)^2 = 100 cm².
Step 7: Now calculate the area: Area = (sqrt(3)/4) * 100 cm² = 25 * sqrt(3) cm².
Step 8: Approximate the value of sqrt(3) (which is about 1.732) to find the area: Area ≈ 25 * 1.732 = 43.3 cm².
Step 9: However, the exact area in terms of sqrt(3) is 25 * sqrt(3) cm², which is the maximum area for the triangle.

Triangle Area Maximization – The area of a triangle is maximized when it is equilateral, given a fixed perimeter.
Perimeter Constraint – Understanding how a fixed perimeter affects the dimensions and area of a triangle.
Geometric Properties – Knowledge of the properties of equilateral triangles and their area formula.