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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A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
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Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
Question
![### Geometry Problem: Circle and Tangents
#### Question 6: Find the value of x.
**Diagram Explanation:**
- The diagram depicts a circle with points R, Q, F, S, and T.
- \( RQ \) is a tangent to the circle at point \( Q \).
- \( S \) is a point on \( RQ \) such that \( \angle RQS = ? \).
- There is a chord \( FT \) that subtends an arc on the circle.
- The given angles in the diagram are:
- \( \angle FQT = 140^\circ \) (an external angle)
- \( \angle FST = 50^\circ \) (an internal angle)
- The task is to find the value of the unknown angle \( x = \angle RQS \).
**Analytical Approach:**
1. As \( RQ \) is a tangent and \( TF \) is a secant to the circle, the tangent-secant angle theorem can be applied here.
2. The angle formed between the tangent at the point of contact and the chord (also extended if necessary) from the point of contact to a point on the circle is half the measure of the intercepted arc on the opposite side of the chord.
**Solution:**
Using the tangent-secant theorem:
\[
\angle RQS = \frac{1}{2} \times \text{intercepted arc}
\]
The intercepted arc in this problem is essentially the internal angle \( 140^\circ \):
\[
\angle RQS = \frac{1}{2} \times 140^\circ = 70^\circ
\]
Thus,
\[
\boxed{\angle RQS = 70^\circ}
\]
This solves the problem of finding the value of \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42779bdc-66e8-47e8-8a59-0cd69b39b8c3%2F6374dc02-4eec-4336-b6d4-055e494ad201%2Fmushiv4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem: Circle and Tangents
#### Question 6: Find the value of x.
**Diagram Explanation:**
- The diagram depicts a circle with points R, Q, F, S, and T.
- \( RQ \) is a tangent to the circle at point \( Q \).
- \( S \) is a point on \( RQ \) such that \( \angle RQS = ? \).
- There is a chord \( FT \) that subtends an arc on the circle.
- The given angles in the diagram are:
- \( \angle FQT = 140^\circ \) (an external angle)
- \( \angle FST = 50^\circ \) (an internal angle)
- The task is to find the value of the unknown angle \( x = \angle RQS \).
**Analytical Approach:**
1. As \( RQ \) is a tangent and \( TF \) is a secant to the circle, the tangent-secant angle theorem can be applied here.
2. The angle formed between the tangent at the point of contact and the chord (also extended if necessary) from the point of contact to a point on the circle is half the measure of the intercepted arc on the opposite side of the chord.
**Solution:**
Using the tangent-secant theorem:
\[
\angle RQS = \frac{1}{2} \times \text{intercepted arc}
\]
The intercepted arc in this problem is essentially the internal angle \( 140^\circ \):
\[
\angle RQS = \frac{1}{2} \times 140^\circ = 70^\circ
\]
Thus,
\[
\boxed{\angle RQS = 70^\circ}
\]
This solves the problem of finding the value of \( x \).
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