In circle O, PQ is a tangent and PRT is a secant. If angle P is 56 degrees and arc QT is 192 degrees, what is the measure of arc QR? (Type in numbers only) 192° 56° P R. T
In circle O, PQ is a tangent and PRT is a secant. If angle P is 56 degrees and arc QT is 192 degrees, what is the measure of arc QR? (Type in numbers only) 192° 56° P R. T
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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![**Title: Understanding Circle Geometry: Tangents and Secants**
**Problem Description:**
In circle O, PQ is a tangent, and PRT is a secant. If angle P measures 56 degrees, and arc QT measures 192 degrees, what is the measure of arc QR? (Type in numbers only)
**Illustration Explanation:**
The given diagram illustrates circle O with several key elements:
- **Circle O**: The center of the circle is labeled as point O.
- **Tangent PQ**: A line segment drawn tangent to the circle at point Q.
- **Secant PRT**: A line segment that intersects the circle at points R and T.
- **Angle P**: Formed by the tangent PQ and the secant PR at the point of tangency Q, measuring 56 degrees.
- **Arc QT**: Part of the circumference of the circle, extending from Q to T, measuring 192 degrees.
**Solution Guide:**
To solve for the measure of arc QR, we use properties of tangents, secants, and arcs in a circle.
1. **Understanding the Angle**: The given angle (56 degrees at point P) between the tangent and the secant is the angle between the tangent line and a chord through the point of tangency.
2. **Arc Relationship**: The angle formed outside the circle (56 degrees) is equal to half the difference of the measures of the intercepted arcs (arc QT and arc QR).
The formula used here is:
\[ \text{Angle} = \frac{1}{2} (\text{measure of major arc} - \text{measure of minor arc}) \]
So, substituting our known values:
\[ 56 = \frac{1}{2} (192 - x) \]
Where \( x \) is the measure of arc QR.
Solve for \( x \):
\[ 56 \times 2 = 192 - x \]
\[ 112 = 192 - x \]
\[ x = 192 - 112 \]
\[ x = 80 \]
Thus, the measure of arc QR is **80 degrees**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccc6c07f-901f-4b9b-9990-bf05e0ed87d5%2Ff4086c95-004d-42ac-8958-4636a412d16e%2Fup0qwb_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Circle Geometry: Tangents and Secants**
**Problem Description:**
In circle O, PQ is a tangent, and PRT is a secant. If angle P measures 56 degrees, and arc QT measures 192 degrees, what is the measure of arc QR? (Type in numbers only)
**Illustration Explanation:**
The given diagram illustrates circle O with several key elements:
- **Circle O**: The center of the circle is labeled as point O.
- **Tangent PQ**: A line segment drawn tangent to the circle at point Q.
- **Secant PRT**: A line segment that intersects the circle at points R and T.
- **Angle P**: Formed by the tangent PQ and the secant PR at the point of tangency Q, measuring 56 degrees.
- **Arc QT**: Part of the circumference of the circle, extending from Q to T, measuring 192 degrees.
**Solution Guide:**
To solve for the measure of arc QR, we use properties of tangents, secants, and arcs in a circle.
1. **Understanding the Angle**: The given angle (56 degrees at point P) between the tangent and the secant is the angle between the tangent line and a chord through the point of tangency.
2. **Arc Relationship**: The angle formed outside the circle (56 degrees) is equal to half the difference of the measures of the intercepted arcs (arc QT and arc QR).
The formula used here is:
\[ \text{Angle} = \frac{1}{2} (\text{measure of major arc} - \text{measure of minor arc}) \]
So, substituting our known values:
\[ 56 = \frac{1}{2} (192 - x) \]
Where \( x \) is the measure of arc QR.
Solve for \( x \):
\[ 56 \times 2 = 192 - x \]
\[ 112 = 192 - x \]
\[ x = 192 - 112 \]
\[ x = 80 \]
Thus, the measure of arc QR is **80 degrees**.
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