Find Circle Area: Perpendicular Radii with Distance y+2

Q: Why do we use the Pythagorean theorem here?

When two radii are perpendicular (meet at 90°), they form a right triangle with the chord AB. The radii are the legs and AB is the hypotenuse, so $ r^2 + r^2 = AB^2 $.

Q: How do I know when radii are perpendicular?

The problem states $ AO \perp OB $, which means the angle between the radii is exactly 90 degrees. You can also look for the small square symbol in diagrams.

Q: Why is the area formula $ \frac{\pi}{2}[y^2+4y+4] $ ?

We found $ r^2 = \frac{y^2+4y+4}{2} $. Since area = $ \pi r^2 $, we get $ \pi \cdot \frac{y^2+4y+4}{2} = \frac{\pi}{2}[y^2+4y+4] $.

Q: Can I factor $ y^2+4y+4 $ ?

Yes! Notice that $ y^2+4y+4 = (y+2)^2 $. This makes sense because AB = y+2, so $ AB^2 = (y+2)^2 $.

Q: What if the radii weren't perpendicular?

If the radii weren't perpendicular, you'd need to use the Law of Cosines instead: $ AB^2 = r^2 + r^2 - 2r^2\cos(\theta) $, where θ is the angle between radii.

Circle Area with Perpendicular Radii

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.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:16 We will express the area of the circle using the variable Y. Let's get started!

00:21 First, we need to find the radius of the circle.

00:26 Now, we'll use the Pythagorean theorem in triangle A, O, B. Are you ready?

00:35 Next, we'll open the parentheses properly. Let's pay attention to each step!

00:51 Here, we isolate R squared. This will help us find the radius.

01:02 Now, we use the formula for calculating the area of a circle.

01:07 Let's substitute the R we just found into the formula. Almost there!

01:19 And that's how we solve this problem. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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.

Step-by-step solution

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

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Right Triangle Rule: When radii are perpendicular, use Pythagorean theorem
Technique: If $AB = y+2$ , then $r^2 + r^2 = (y+2)^2$
Check: Substitute back: if $r^2 = \frac{y^2+4y+4}{2}$ , then $2r^2 = (y+2)^2$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to apply Pythagorean theorem correctly
Don't treat perpendicular radii like they form a straight line = missing the right triangle! This leads to setting AB = 2r instead of using $r^2 + r^2 = AB^2$ . Always recognize that perpendicular radii create a right triangle with the chord as hypotenuse.

Practice Quiz

Test your knowledge with interactive questions

Consider a right-angled triangle, AB = 8 cm and AC = 6 cm.
Calculate the length of side BC.

FAQ

Everything you need to know about this question

Why do we use the Pythagorean theorem here?

When two radii are perpendicular (meet at 90°), they form a right triangle with the chord AB. The radii are the legs and AB is the hypotenuse, so $r^2 + r^2 = AB^2$ .

How do I know when radii are perpendicular?

The problem states $AO \perp OB$ , which means the angle between the radii is exactly 90 degrees. You can also look for the small square symbol in diagrams.

Why is the area formula $\frac{\pi}{2}[y^2+4y+4]$ ?

We found $r^2 = \frac{y^2+4y+4}{2}$ . Since area = $\pi r^2$ , we get $\pi \cdot \frac{y^2+4y+4}{2} = \frac{\pi}{2}[y^2+4y+4]$ .

Can I factor $y^2+4y+4$ ?

Yes! Notice that $y^2+4y+4 = (y+2)^2$ . This makes sense because AB = y+2, so $AB^2 = (y+2)^2$ .

What if the radii weren't perpendicular?

If the radii weren't perpendicular, you'd need to use the Law of Cosines instead: $AB^2 = r^2 + r^2 - 2r^2\cos(\theta)$ , where θ is the angle between radii.

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Find Circle Area: Perpendicular Radii with Distance y+2

Circle Area with Perpendicular Radii

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we use the Pythagorean theorem here?

How do I know when radii are perpendicular?

Why is the area formula $\frac{\pi}{2}[y^2+4y+4]$ ?

Can I factor $y^2+4y+4$ ?

What if the radii weren't perpendicular?

🌟 Unlock Your Math Potential

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❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we use the Pythagorean theorem here?

How do I know when radii are perpendicular?

Why is the area formula π2[y2+4y+4] \frac{\pi}{2}[y^2+4y+4] 2π​[y2+4y+4]?

Can I factor y2+4y+4 y^2+4y+4 y2+4y+4?

What if the radii weren't perpendicular?

🌟 Unlock Your Math Potential

More Questions

Square of sum

Solve the System: x²+4=-6y and y²+9=-4x Equations

Solve Rational Expression: (2x+1)²/(x+2) + (x+2)²/(2x+1) = 4.5x

Calculate A and B in the Equation: Solving \((\sqrt{x}-\sqrt{x+1})/(x+1)=1\)

Evaluate the Expression: What is (a+3b)²-(3b-a)²?

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Compare Circle Areas: Finding the Ratio When r₁ = 6cm and d₂ = 12cm

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Why is the area formula $\frac{\pi}{2}[y^2+4y+4]$ ?

Can I factor $y^2+4y+4$ ?