https://file.aiquickdraw.com/mathbot/file/1725482374066z9t2jkyo.jpg

The image shows a geometry problem involving a sector of a circle. The task is to solve two parts of a problem related to the sector.

### Given:
- The area of the sector \( AOB \) is \( 25 \, \text{cm}^2 \).
- The length \( AB \) is required.

### Tasks:
1. **Express the radius** of the sector.
2. **Find the length** of the arc \( AB \).

### Solution:

Let's denote:
- \( r \) as the radius of the circle.
- \( \theta \) as the central angle \( \angle AOB \) in radians.

#### Step 1: Express the Radius

The area of the sector is given by:

\[
\text{Area of Sector} = \frac{1}{2} r^2 \theta
\]

Given:

\[
\frac{1}{2} r^2 \theta = 25 \, \text{cm}^2
\]

Therefore,

\[
r^2 \theta = 50 \, \text{cm}^2
\]

Thus, the expression for the radius \( r \) in terms of \( \theta \) is:

\[
r = \sqrt{\frac{50}{\theta}} \, \text{cm}
\]

#### Step 2: Find the Length of Arc \( AB \)

The length of the arc \( AB \) is given by:

\[
\text{Length of arc} = r \theta
\]

Using the expression for \( r \) from Step 1:

\[
\text{Length of arc} = \sqrt{\frac{50}{\theta}} \cdot \theta = \theta \sqrt{\frac{50}{\theta}} = \sqrt{50 \theta} \, \text{cm}
\]

### Final Answers:
1. The radius \( r \) is expressed as \( r = \sqrt{\frac{50}{\theta}} \) cm.
2. The length of the arc \( AB \) is \( \sqrt{50 \theta} \) cm.

Would you like further details, or do you have any questions?

---

Here are five related questions to consider:

1. How do you convert an angle in degrees to radians?
2. What would happen to the length of the arc if the angle \( \theta \) were doubled?
3. How does changing the radius \( r \) affect the area of the sector?
4. How can you find the central angle \( \theta \) if both the area and the radius are known?
5. What is the relationship between the arc length and the radius?

**Tip:** Remember that radians are often more useful than degrees in trigonometric and calculus contexts, especially when working with arc lengths and sector areas.

The area of the sector AOB is 25 cm^2. (i) Express the radius. (ii) Find the length of arc AB.

Solve a geometry problem involving a sector with an area of 25 cm². Learn how to express the radius and find the length of the arc AB using the sector area formula and the arc length formula. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation