1. Priya drew a circle. Han wants to draw a circle with a circumference 4 times the circumference of Priya's circle. Han thinks his circle should have a radius 4 times the radius of Priya's circle. Do you agree with Han?

Yes, I agree with Han's reasoning. Here's why: ### Understanding the Relationship Between Circumference and Radius The **circumference** \( C \) of a circle is directly proportional to its **radius** \( r \). This relationship is given by the formula: \[ C = 2\pi r \] ### Priya's Circle Let's denote: - **Priya's radius** as \( r \) - **Priya's circumference** as \( C = 2\pi r \) ### Han's Desired Circle Han wants his circle to have a circumference that is **4 times** Priya's circumference: \[ C' = 4C = 4 \times 2\pi r = 8\pi r \] Using the circumference formula for Han's circle: \[ C' = 2\pi R \] where \( R \) is Han's radius. Setting the two expressions for \( C' \) equal to each other: \[ 2\pi R = 8\pi r \] Solving for \( R \): \[ R = \frac{8\pi r}{2\pi} = 4r \] ### Conclusion Since Han's radius \( R \) is **4 times** Priya's radius \( r \), his calculation correctly achieves a circumference that is 4 times larger than Priya's. Therefore, Han's approach and reasoning are **correct**.