Question

Find the area (in squared units) of the largest trapezoid that can be inscribed in a circle of radius 5 units and whose base is a diameter of the circle.

Answer 1

The area of the largest trapezoid that can be inscribed in a circle with radius 5 units, where the base is a diameter of the circle, is 50 square units. This trapezoid is an isosceles trapezoid, and the calculation is based on trapezoid area formulas and properties of circles.

Step-by-step explanation:

To find the area of the largest trapezoid that can be inscribed in a circle of radius 5 units, with one of its bases being a diameter of the circle, we'll need to use some geometric principles and properties of trapezoids and circles.

Since one of the bases of the trapezoid is the diameter of the circle, this base is of length 10 units because the diameter is twice the radius. The other base and the legs of the trapezoid are tangent to the circle. This kind of trapezoid is called an isosceles trapezoid because the legs (the non-parallel sides) are equal in length.

The area (A) of a trapezoid can be found using the formula:

A = (1/2) * (base1 + base2) * height

For the largest trapezoid, the height will be equal to the radius of the circle (5 units), as it can be seen as the altitude from the center to the base that isn't the diameter. Thus, we can calculate the base2 using the properties of the right triangle formed by the height, half of base2, and the radius.

Let's designate half the length of the unknown base as x. Then, using the Pythagorean theorem for one of the right triangles we have:

52 = x2 + 52 => x = 5

Now, base2 = 2x = 10 units, and we have an isosceles trapezoid with both bases 10 units in length. The area of such a trapezoid is:

A = (1/2) * (10 + 10) * 5 = 50 square units.

Answer 2

The area of the largest trapezoid that can be inscribed in a circle of radius 5 units, with its base as the diameter of the circle, is 50 square units.

Step-by-step explanation:

Finding the area of the largest trapezoid inscribed in a circle involves understanding that the larger base of the trapezoid will be the diameter of the circle, and the legs will be tangent to the circle. Since the diameter is the longest chord of the circle, it will also be the longest possible base for the trapezoid inside the circle. The height of the trapezoid will be equal to the radius of the circle because if we draw radii to the points where the legs touch the circle, they will form a right angle with the legs, creating a pair of right triangles within the trapezoid.

Given a circle of radius 5 units, its diameter will be 10 units. To find the area of a trapezoid, we use the formula A = ((b1 + b₂) / 2) × h, where b1 and b₂ are the lengths of the bases and h is the height. In this case, b₁ is the diameter (10 units), and b₂ will also be the diameter since it's the largest trapezoid that can be inscribed in the circle. The height h is the radius of the circle, which is 5 units. Therefore, the area A of the trapezoid is ((10 + 10) / 2) × 5 = 50 square units.