Find Circle Area: Perpendicular Radii with Distance y+2

Question

Given a circle whose center O. From the center of the circle go out 2 radii that cut the circle at the points A and B.

Given AO⊥OB.

The side AB is equal to and+2.

Express band and the area of the circle.

00:00 Express the area of the circle using Y

00:03 Circle radius

00:10 Let's use the Pythagorean theorem in triangle AOB

00:19 Let's properly open the parentheses

00:35 Let's isolate R squared

00:46 Let's use the formula for calculating circle area

00:49 Let's substitute the R we just found

01:03 And this is the solution to the question

To solve this problem, we'll follow these steps:

Identify the given information.
Use the geometric properties of a circle and a right triangle to find the radius.
Express the area of the circle in terms of the given expression.

Now, let's work through each step:

Step 1: Given a circle with center $O$ and radii $AO$ and $OB$ such that $AO\perp OB$ , each is a radius $r$ , and $AB = \text{and}+2$ .

Step 2: By the Pythagorean theorem, we know:

AO^2 + OB^2 = AB^2

r^2 + r^2 = (y+2)^2

2r^2 = y^2 + 4y + 4

Step 3: Solving for the area of the circle:

The radius $r$ can be expressed by rearranging:

r^2 = \frac{y^2 + 4y + 4}{2}

The area of the circle using this radius is:

\text{Area} = \pi r^2 = \pi \left(\frac{y^2 + 4y + 4}{2}\right) = \frac{\pi}{2}(y^2 + 4y + 4)

Therefore, the expression for the area of the circle is $\frac{\pi}{2}[y^2+4y+4]$ .

$\frac{\pi}{2}[y^2+4y+4]$