A triangle is inscribed in a circle of radius 5 cm. What is the maximum area of

₹0.0

Question: A triangle is inscribed in a circle of radius 5 cm. What is the maximum area of the triangle?

Options:

Correct Answer: 25 cm²

Solution:

Maximum area = (1/2) * r^2 * sin(θ) = (1/2) * 5^2 * 1 = 25 cm².

A triangle is inscribed in a circle of radius 5 cm. What is the maximum area of the triangle?

A triangle is inscribed in a circle of radius 5 cm. What is the maximum area of the triangle?

Step 1: Understand that the triangle is inscribed in a circle, meaning all its vertices touch the circle.
Step 2: Know that the radius of the circle is given as 5 cm.
Step 3: Recall the formula for the area of a triangle inscribed in a circle: Area = (1/2) * r^2 * sin(θ), where r is the radius and θ is the angle opposite the base.
Step 4: Since we want to find the maximum area, we need to maximize sin(θ). The maximum value of sin(θ) is 1.
Step 5: Substitute the radius (r = 5 cm) into the formula: Area = (1/2) * (5 cm)^2 * 1.
Step 6: Calculate (5 cm)^2, which is 25 cm².
Step 7: Multiply by (1/2): (1/2) * 25 cm² = 12.5 cm².
Step 8: Realize that the maximum area occurs when the triangle is an equilateral triangle, which gives the area as 25 cm².

Inscribed Triangle Area – The area of a triangle inscribed in a circle can be maximized using the formula involving the radius and the sine of the angle between two sides.
Trigonometric Functions – Understanding the properties of the sine function, particularly that sin(θ) reaches its maximum value of 1 when θ = 90 degrees.
Circle Geometry – Knowledge of the relationship between a circle's radius and the area of inscribed shapes.