https://file.aiquickdraw.com/mathbot/file/17286120920946lgjmyxi.jpg

In the given diagram, line \( TQ \) is tangent to a circle with center \( S \) at point \( Q \), and chord \( RQ \) is a diameter of the circle. We are asked to find \( m \angle PQT \), given that \( x = 28^\circ \).

### Key points:
1. \( TQ \) is a tangent to the circle at point \( Q \).
2. \( RQ \) is a diameter of the circle.
3. The angle between a tangent and a chord through the point of tangency is equal to the inscribed angle on the opposite side of the chord.

This means that \( m \angle PQT = m \angle PRQ \) (since they are both subtended by the same arc \( PQ \)).

### Solution:
- We know that \( x = 28^\circ \).
- The inscribed angle \( \angle PRQ \) subtended by the diameter of a circle is a right angle, meaning \( \angle PRQ = 90^\circ \).

Using the property of angles around point \( S \), we know:
\[
m \angle PRQ = m \angle PQT = 90^\circ - x
\]
Substituting \( x = 28^\circ \):
\[
m \angle PQT = 90^\circ - 28^\circ = 62^\circ
\]

Thus, the correct answer is \( \boxed{62^\circ} \) (Option A).

Let me know if you would like further details or have any questions!

---

### Related Questions:
1. What is the relationship between a tangent and the radius at the point of tangency?
2. How can you determine the measure of an angle subtended by a chord?
3. What is the property of the angles subtended by a diameter of a circle?
4. How does the exterior angle theorem apply in this context?
5. How would the result change if \( x \) had a different value?

### Tip:
Always remember that the angle formed by a tangent and a chord through the point of tangency is equal to the angle subtended by the same chord on the opposite side of the circle.

In the given diagram, line TQ is tangent to a circle with center S at point Q, and chord RQ is a diameter of the circle. If x = 28°, what is m∠PQT?

Solve the geometry problem involving a tangent to a circle and a diameter. Given that x = 28°, learn how to calculate m∠PQT using key circle theorems and the inscribed angle formula. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation