Two circles intersect each other at the points A and B, A straight line parallel to AB intersects the circles at C, D, E and F. If CD = 4.5 cm, then the measure of EF is

Options

A. `1.50 cm `
B. `2.25 cm `
C. `4.50 cm `
D. `9.00 cm `

Answer

To solve the problem, we need to find the length of EF given that CD = 4.5 cm and that the line segment CD is parallel to the line segment EF.

### Step-by-step Solution:

1. **Understanding the Configuration**:
- We have two circles that intersect at points A and B.
- A line parallel to AB intersects the circles at points C, D, E, and F.

2. **Given Information**:
- The length of segment CD is given as 4.5 cm.

3. **Properties of Parallel Lines**:
- Since CD is parallel to EF, and both segments are intercepted by the same two circles, we can use the property of parallel lines. This property states that if two lines are parallel and are cut by a transversal (in this case, the line connecting the points of intersection), the segments formed on the circles will be proportional.

4. **Using the Perpendicular from the Centers**:
- Let’s denote the centers of the circles as O1 and O2.
- If we drop perpendiculars from O1 and O2 to the line segments CD and EF, we can denote the points where these perpendiculars meet the segments as N (for CD) and M (for EF).
- By the properties of circles, the segments will be equal because the perpendicular from the center to a chord bisects the chord.

5. **Setting Up the Equation**:
- Since CN = NF and DN = NE (because of the perpendiculars), we can express the lengths as:
- CD = CN + DN
- EF = NE + NF
- Since CN = NF and DN = NE, we can say:
- CD = 2 * CN
- EF = 2 * NE

6. **Finding the Length of EF**:
- Since CD = 4.5 cm, we can conclude that:
- EF = CD
- Thus, EF = 4.5 cm.

### Conclusion:
The measure of EF is **4.5 cm**.