Question

A rectangle whose base is twice it’s altitude is inscribed in a circle whose radius is 5mm. Find the area of the rectangle.

Answer 1

Answer and Explanation:

40 square mm

See the attached image.

Basically, the 5mm radius that is given is half the diagonal of the rectangle. So if we double the radius, it becomes the diagonal, which is the hypotenuse of the right triangle that can be constructed of half the rectangle. From there, we can use the Pythagorean theorem to solve for the rectangle's side length by substituting a for
`\ell` and b for
`2\ell`. Finally, we multiply
`\ell` and
`2\ell` to get the area of the rectangle:

`A = \ell \cdot 2\ell`

`A = 2\sqrt5 \cdot 2(2 √(5))`

`A = 2^3 \cdot (\sqrt5)^2`

`A = 8 \cdot 5`

`A = 40 \, \textrm{mm}^2`