Square of sum: Using additional geometric shapes

Remember that the area of a square is equal to the side of the square squared.

The formula for the area of a square is:

$S=a^2$

Finally, substitute the data into the formula:

$S=(5+2x)^2$

To find $x$ in the given triangle, let's apply the Pythagorean Theorem. The squared lengths of the triangle's legs and hypotenuse are related by this equation:

First, expand each term:

• $(x+1)^2 = x^2 + 2x + 1$
• $x^2 = x^2$
• $(x+2)^2 = x^2 + 4x + 4$

Plug these into the Pythagorean Theorem equation:

Combine like terms:

Rearrange the equation to isolate terms on one side:

Simplify to get a quadratic equation:

Now, solve for $x$ using factoring. Look for two numbers that multiply to $-3$ and add to $-2$. These numbers are $-3$ and $1$:

Set each factor equal to zero:

• $x - 3 = 0 \Rightarrow x = 3$
• $x + 1 = 0 \Rightarrow x = -1$

Given the condition $x > 0$, the valid solution is:

$x = 3$

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:

Step 3: Solving for the area of the circle:

The radius $r$ can be expressed by rearranging:

The area of the circle using this radius is:

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