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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Chord KO and secant PL intersect at N. What is m
![### Geometry Problem – Chord and Secant Intersection
**Question:**
4. Chord KO and secant PL intersect at N. What is the measure of angle ∠KNL?
- (A) 20°
- (B) 30°
- (C) 60°
- (D) 80°
#### Diagram Explanation:
The accompanying diagram illustrates a circle with center O. In the diagram:
1. A chord KO intersects a secant PL at point N.
2. ∠KML is marked as 60°.
3. ∠PKO is marked as 100°.
4. Points K, L, P, and O are placed approximately as follows:
- K is on the circumference near the top-right.
- L is on the extended line of PL to the top-right.
- P is on the extended line of PL toward the bottom-left.
- N is the point of intersection of KO and PL inside the circle.
### Solution
To find the measure of ∠KNL, recognizing the angles formed and the specific properties of the circle's chords and secants is essential. One key property of circle intersections involving chords and secants is helpful: the measure of an angle formed by two intersecting chords (or a chord and secant) is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Given:
- ∠KML (intercepted arc by angle ∠KNL) = 60°.
- ∠PKO (intercepted arc by the vertical angle point) = 100°.
Calculating,
\[ \text{∠KNL} = \frac{1}{2} \times (\text{intercepted arc measurements}) \]
\[ \text{∠KNL} = \frac{1}{2} \times (100° + 60°) \]
\[ \text{∠KNL} = \frac{1}{2} \times 160° \]
\[ \text{∠KNL} = 80° \]
**Answer:** (D) 80°](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F136eda30-b130-471a-a70d-0d5186fbd89f%2F2d905105-6190-44f2-8afc-f755d7e1ffaa%2F4sj2zi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem – Chord and Secant Intersection
**Question:**
4. Chord KO and secant PL intersect at N. What is the measure of angle ∠KNL?
- (A) 20°
- (B) 30°
- (C) 60°
- (D) 80°
#### Diagram Explanation:
The accompanying diagram illustrates a circle with center O. In the diagram:
1. A chord KO intersects a secant PL at point N.
2. ∠KML is marked as 60°.
3. ∠PKO is marked as 100°.
4. Points K, L, P, and O are placed approximately as follows:
- K is on the circumference near the top-right.
- L is on the extended line of PL to the top-right.
- P is on the extended line of PL toward the bottom-left.
- N is the point of intersection of KO and PL inside the circle.
### Solution
To find the measure of ∠KNL, recognizing the angles formed and the specific properties of the circle's chords and secants is essential. One key property of circle intersections involving chords and secants is helpful: the measure of an angle formed by two intersecting chords (or a chord and secant) is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Given:
- ∠KML (intercepted arc by angle ∠KNL) = 60°.
- ∠PKO (intercepted arc by the vertical angle point) = 100°.
Calculating,
\[ \text{∠KNL} = \frac{1}{2} \times (\text{intercepted arc measurements}) \]
\[ \text{∠KNL} = \frac{1}{2} \times (100° + 60°) \]
\[ \text{∠KNL} = \frac{1}{2} \times 160° \]
\[ \text{∠KNL} = 80° \]
**Answer:** (D) 80°
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