(C) Your students have learned that if two chords intersect within a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord. That is, in the following diagram, ab = cd. Some of your more curious students want to know how that is proven. How do you do it? [Hint: Draw the dotted lines as shown in Figure 5.32, and try to get similar triangles.] R O b Figure 5.32 'S
(C) Your students have learned that if two chords intersect within a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord. That is, in the following diagram, ab = cd. Some of your more curious students want to know how that is proven. How do you do it? [Hint: Draw the dotted lines as shown in Figure 5.32, and try to get similar triangles.] R O b Figure 5.32 'S
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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![(C) Your students have learned that if two chords intersect within a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord. That is, in the following diagram, \( ab = cd \). Some of your more curious students want to know how that is proven. How do you do it? [Hint: Draw the dotted lines as shown in Figure 5.32, and try to get similar triangles.]
**Figure 5.32 Explanation:**
- The circle contains an intersecting system of chords.
- Chord \( PQ \) is intersected by chord \( RS \) inside the circle at point \( T \).
- The segments of the chords are labeled as follows:
- \( PT = a \)
- \( TQ = b \)
- \( RT = c \)
- \( TS = d \)
- According to the intersecting chords theorem, \( ab = cd \).
To understand this theorem, it is helpful to draw the additional lines connecting \( P \) to \( S \) and \( Q \) to \( R \) as hinted. This creates triangles \( \triangle PTR \), \( \triangle QTS \), \( \triangle QTR \), \( \triangle PTS \), which can help in finding similar triangles and proving the theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98b6e310-08ba-4e1d-a9bc-704b45d2ce6c%2F4673196e-3a5f-4ed4-bd4d-641a24b28100%2Fybl3rty_processed.png&w=3840&q=75)
Transcribed Image Text:(C) Your students have learned that if two chords intersect within a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord. That is, in the following diagram, \( ab = cd \). Some of your more curious students want to know how that is proven. How do you do it? [Hint: Draw the dotted lines as shown in Figure 5.32, and try to get similar triangles.]
**Figure 5.32 Explanation:**
- The circle contains an intersecting system of chords.
- Chord \( PQ \) is intersected by chord \( RS \) inside the circle at point \( T \).
- The segments of the chords are labeled as follows:
- \( PT = a \)
- \( TQ = b \)
- \( RT = c \)
- \( TS = d \)
- According to the intersecting chords theorem, \( ab = cd \).
To understand this theorem, it is helpful to draw the additional lines connecting \( P \) to \( S \) and \( Q \) to \( R \) as hinted. This creates triangles \( \triangle PTR \), \( \triangle QTS \), \( \triangle QTR \), \( \triangle PTS \), which can help in finding similar triangles and proving the theorem.
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