Question
An interior designer is hanging a circular clock for a client, as shown. The hanger at point B connects to the clock by two wires that are tangent to the clock at points A and C.
A circle is shown with center at point E. There is a line segment connecting points B, D, E, and F. Segments DE and EF are radii of the circle. Segment DF is a diameter of the circle. Segment AB and BC are tangent to the circle at points A and C.
If the radius of the clock is 20 cm and the distance from the top of the clock at point D to the hanger at point B is 5 cm, what is the length from point A to point B?
5 cm
15 cm
20 cm
45 cm
A circle is shown with center at point E. There is a line segment connecting points B, D, E, and F. Segments DE and EF are radii of the circle. Segment DF is a diameter of the circle. Segment AB and BC are tangent to the circle at points A and C.
If the radius of the clock is 20 cm and the distance from the top of the clock at point D to the hanger at point B is 5 cm, what is the length from point A to point B?
5 cm
15 cm
20 cm
45 cm
Answers
GPT-4o mini
To solve for the length from point A to point B, we first need to analyze the situation using geometry.
1. **Identify the known values**:
- The radius of the clock (circle) is 20 cm. Therefore, the distance from the center E of the clock to points A and C (where the tangents touch the circle) is 20 cm.
- The distance from the top of the clock at point D to the hanger at point B is 5 cm.
2. **Determine the lengths from the center E to point D and point B**:
- The distance from the center E to point D (the top of the clock) is also equal to the radius of the clock, which is 20 cm.
- Therefore, the total distance from point E to point B can be calculated as:
\[
\text{Distance from E to B} = \text{Distance from E to D} + \text{Distance from D to B} = 20 \text{ cm} + 5 \text{ cm} = 25 \text{ cm}
\]
3. **Using right triangle properties**:
- Since segments AB and BC are tangents to the circle at points A and C, triangle EAB is a right triangle.
- In right triangle EAB, we have:
- EB (hypotenuse) = 25 cm (calculated above)
- EA (radius, one leg) = 20 cm
4. **Applying the Pythagorean theorem**:
To find AB (the tangent segment), we can use:
\[
EB^2 = EA^2 + AB^2
\]
Plugging in the values we know:
\[
25^2 = 20^2 + AB^2
\]
This simplifies to:
\[
625 = 400 + AB^2
\]
\[
AB^2 = 625 - 400 = 225
\]
\[
AB = \sqrt{225} = 15 \text{ cm}
\]
5. **Conclusion**:
Therefore, the length from point A to point B is:
\[
\boxed{15 \text{ cm}}
\]
1. **Identify the known values**:
- The radius of the clock (circle) is 20 cm. Therefore, the distance from the center E of the clock to points A and C (where the tangents touch the circle) is 20 cm.
- The distance from the top of the clock at point D to the hanger at point B is 5 cm.
2. **Determine the lengths from the center E to point D and point B**:
- The distance from the center E to point D (the top of the clock) is also equal to the radius of the clock, which is 20 cm.
- Therefore, the total distance from point E to point B can be calculated as:
\[
\text{Distance from E to B} = \text{Distance from E to D} + \text{Distance from D to B} = 20 \text{ cm} + 5 \text{ cm} = 25 \text{ cm}
\]
3. **Using right triangle properties**:
- Since segments AB and BC are tangents to the circle at points A and C, triangle EAB is a right triangle.
- In right triangle EAB, we have:
- EB (hypotenuse) = 25 cm (calculated above)
- EA (radius, one leg) = 20 cm
4. **Applying the Pythagorean theorem**:
To find AB (the tangent segment), we can use:
\[
EB^2 = EA^2 + AB^2
\]
Plugging in the values we know:
\[
25^2 = 20^2 + AB^2
\]
This simplifies to:
\[
625 = 400 + AB^2
\]
\[
AB^2 = 625 - 400 = 225
\]
\[
AB = \sqrt{225} = 15 \text{ cm}
\]
5. **Conclusion**:
Therefore, the length from point A to point B is:
\[
\boxed{15 \text{ cm}}
\]