Circles With Centers A And B Have Radii 4 And 9, Respectively. A Common Internal Tangent Touches The Circles At C And D as Shown. Lines AB And CD Intersect At E And AE 6. What Is The Length Of CD? Give Your Answer As A Decimal number To The Nearest
Introduction
In geometry, circles are a fundamental concept that has been studied for centuries. A circle is a set of points that are all equidistant from a central point called the center. The distance from the center to any point on the circle is called the radius. In this article, we will explore a problem involving two circles with centers A and B, and radii 4 and 9, respectively. A common internal tangent touches the circles at C and D, and lines AB and CD intersect at E. Given that AE = 6, we will find the length of CD.
Understanding the Problem
To solve this problem, we need to understand the geometric properties of circles and tangents. A tangent to a circle is a line that touches the circle at exactly one point. In this case, the common internal tangent touches the circles at C and D. Since the tangent is internal, it lies inside the circles and intersects the line segment AB at E.
Drawing the Diagram
To visualize the problem, let's draw a diagram. We will draw two circles with centers A and B, and radii 4 and 9, respectively. We will also draw the common internal tangent, which touches the circles at C and D. Finally, we will draw the line segment AB and the point E where the tangent intersects AB.
+---------------+
| |
| Circle B |
| (radius = 9) |
| |
+---------------+
|
|
v
+---------------+
| |
| Circle A |
| (radius = 4) |
| |
+---------------+
|
|
v
+---------------+
| |
| Tangent |
| (touches C |
| and D) |
| |
+---------------+
|
|
v
+---------------+
| |
| Line AB |
| (AE = 6) |
| |
+---------------+
|
|
v
+---------------+
| |
| Point E |
| |
+---------------+
Finding the Length of CD
To find the length of CD, we need to use the properties of similar triangles. Since the tangent is internal, the angles at C and D are equal. Therefore, the triangles ACE and BDE are similar.
Similar Triangles
Two triangles are similar if their corresponding angles are equal. In this case, the triangles ACE and BDE are similar because they have equal angles at C and D.
Using Similar Triangles
Since the triangles ACE and BDE are similar, we can set up a proportion to find the length of CD. Let's call the length of CD x. Then, we can set up the following proportion:
(AE / AC) = (BE / BD)
(6 / 4) = (BE / (9 - x))
Solving for x
To solve for x, we can cross-multiply and simplify the equation:
6(9 - x) = 4BE
54 - 6x = 4BE
54 - 6x = 4(9 - x)
54 - 6x = 36 - 4x
18 = 2x
x = 9
Conclusion
In this article, we used the properties of similar triangles to find the length of CD. We drew a diagram to visualize the problem and set up a proportion to find the length of CD. By solving for x, we found that the length of CD is 9.
Final Answer
Introduction
In our previous article, we explored a problem involving two circles with centers A and B, and radii 4 and 9, respectively. A common internal tangent touches the circles at C and D, and lines AB and CD intersect at E. Given that AE = 6, we found the length of CD to be 9. In this article, we will answer some frequently asked questions related to this problem.
Q: What is the significance of the internal tangent?
A: The internal tangent is a line that touches the circles at exactly one point. In this case, the internal tangent touches the circles at C and D. The internal tangent is significant because it helps us to find the length of CD.
Q: Why are the triangles ACE and BDE similar?
A: The triangles ACE and BDE are similar because they have equal angles at C and D. Since the tangent is internal, the angles at C and D are equal, which makes the triangles similar.
Q: How do we use similar triangles to find the length of CD?
A: We use similar triangles to set up a proportion to find the length of CD. Let's call the length of CD x. Then, we can set up the following proportion:
(AE / AC) = (BE / BD)
(6 / 4) = (BE / (9 - x))
Q: What is the significance of the proportion (AE / AC) = (BE / BD)?
A: The proportion (AE / AC) = (BE / BD) is significant because it helps us to find the length of CD. By setting up this proportion, we can solve for x, which is the length of CD.
Q: How do we solve for x in the proportion (AE / AC) = (BE / BD)?
A: To solve for x, we can cross-multiply and simplify the equation:
6(9 - x) = 4BE
54 - 6x = 4BE
54 - 6x = 4(9 - x)
54 - 6x = 36 - 4x
18 = 2x
x = 9
Q: What is the final answer for the length of CD?
A: The final answer for the length of CD is 9.
Q: What are some real-world applications of this problem?
A: This problem has several real-world applications, such as:
- Architecture: In architecture, circles and tangents are used to design buildings and structures. Understanding the properties of circles and tangents is essential for designing stable and aesthetically pleasing structures.
- Engineering: In engineering, circles and tangents are used to design machines and mechanisms. Understanding the properties of circles and tangents is essential for designing efficient and reliable machines.
- Computer Science: In computer science, circles and tangents are used to design algorithms and data structures. Understanding the properties of circles and tangents is essential for designing efficient and scalable algorithms.
Conclusion
In this article, we answered some frequently asked questions related to the problem of circles with centers A and B. We discussed the significance of the internal tangent, the similarity of triangles ACE and BDE, and how to use similar triangles to find the length of CD. We also discussed some real-world applications of this problem.