Solve for x. 27

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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### Solve for \( x \)

The diagram presented is a circle with two line segments intersecting inside it.

#### Diagram Description:
- The circle contains two intersecting chords.
- The first chord is divided into two segments by the intersection point, with one segment measuring 9 units, and the other, labeled as \( x \), is unknown.
- The second chord is similarly divided into two segments, with one segment measuring 9 units and the other segment measuring 27 units.

#### Solution Explanation:
To solve for \( x \), we use the Intersecting Chords Theorem, which states that if two chords intersect inside a circle, the products of the lengths of their segments are equal.

Mathematically, the theorem can be expressed as:
\[ (segment_1 \times segment_2)_{chord1} = (segment_1 \times segment_2)_{chord2} \]

From the diagram:
- For the first chord, the segments are 9 and \( x \).
- For the second chord, the segments are 9 and 27.

Applying the theorem:
\[ 9 \times x = 9 \times 27 \]

Simplifying the equation:
\[ x = 27 \]

Thus, \( x \) is 27 units.
Transcribed Image Text:### Solve for \( x \) The diagram presented is a circle with two line segments intersecting inside it. #### Diagram Description: - The circle contains two intersecting chords. - The first chord is divided into two segments by the intersection point, with one segment measuring 9 units, and the other, labeled as \( x \), is unknown. - The second chord is similarly divided into two segments, with one segment measuring 9 units and the other segment measuring 27 units. #### Solution Explanation: To solve for \( x \), we use the Intersecting Chords Theorem, which states that if two chords intersect inside a circle, the products of the lengths of their segments are equal. Mathematically, the theorem can be expressed as: \[ (segment_1 \times segment_2)_{chord1} = (segment_1 \times segment_2)_{chord2} \] From the diagram: - For the first chord, the segments are 9 and \( x \). - For the second chord, the segments are 9 and 27. Applying the theorem: \[ 9 \times x = 9 \times 27 \] Simplifying the equation: \[ x = 27 \] Thus, \( x \) is 27 units.
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