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8. In the diagram below, circles A and B are tangent at point C and AB is drawn. Sketch all common tangent lines.

8. In the diagram below, circles A and B are tangent at point C and AB is drawn. Sketch all common tangent lines.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Problem 8: Tangent Circles and Common Tangents** *Question:* In the diagram below, circles \( A \) and \( B \) are tangent at point \( C \) and \( \overline{AB} \) is drawn. Sketch all common tangent lines. *Diagram Description:* The diagram consists of two circles that are tangent to each other at a single point, denoted as \( C \). Circle \( A \) is larger and circle \( B \) is smaller. The line segment \( \overline{AB} \) is drawn, connecting the centers of the two circles, passing through the point of tangency \( C \). *Explanation:* When two circles are tangent to each other at a single point, they share a point of tangency, and the centers of the circles, together with the point of tangency, lie along a straight line. Common tangents to both circles can be classified into two categories: 1. **External Tangents**: These tangents do not pass between the circles but touch each circle at one point outside the area between them. 2. **Internal Tangents**: These tangents pass between the circles and touch each circle at one point inside the area between them. In this case, common tangents will be found as follows: - **External Tangents**: 1. Draw a line that touches both circles at a point outside the area between them. 2. There will be two such tangents, one above and one below the line \( \overline{AB} \). - **Internal Tangent**: 1. Draw a line that touches both circles at a point inside the area between them, along the line \( \overline{AB} \). In practice, to sketch this, you would: 1. Draw the two circles with their respective radii so that they are tangent at point \( C \). 2. Draw \( \overline{AB} \), the line segment connecting the centers of circle \( A \) and circle \( B \), passing through \( C \). 3. From points outside the area between the circles, draw two external tangents touching each circle. 4. For the internal tangent, draw the line along \( \overline{AB} \) and identify the single touching point on each circle. By identifying and drawing these tangents, you illustrate the common tangents between the two tangent circles
Transcribed Image Text:**Problem 8: Tangent Circles and Common Tangents** *Question:* In the diagram below, circles \( A \) and \( B \) are tangent at point \( C \) and \( \overline{AB} \) is drawn. Sketch all common tangent lines. *Diagram Description:* The diagram consists of two circles that are tangent to each other at a single point, denoted as \( C \). Circle \( A \) is larger and circle \( B \) is smaller. The line segment \( \overline{AB} \) is drawn, connecting the centers of the two circles, passing through the point of tangency \( C \). *Explanation:* When two circles are tangent to each other at a single point, they share a point of tangency, and the centers of the circles, together with the point of tangency, lie along a straight line. Common tangents to both circles can be classified into two categories: 1. **External Tangents**: These tangents do not pass between the circles but touch each circle at one point outside the area between them. 2. **Internal Tangents**: These tangents pass between the circles and touch each circle at one point inside the area between them. In this case, common tangents will be found as follows: - **External Tangents**: 1. Draw a line that touches both circles at a point outside the area between them. 2. There will be two such tangents, one above and one below the line \( \overline{AB} \). - **Internal Tangent**: 1. Draw a line that touches both circles at a point inside the area between them, along the line \( \overline{AB} \). In practice, to sketch this, you would: 1. Draw the two circles with their respective radii so that they are tangent at point \( C \). 2. Draw \( \overline{AB} \), the line segment connecting the centers of circle \( A \) and circle \( B \), passing through \( C \). 3. From points outside the area between the circles, draw two external tangents touching each circle. 4. For the internal tangent, draw the line along \( \overline{AB} \) and identify the single touching point on each circle. By identifying and drawing these tangents, you illustrate the common tangents between the two tangent circles
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