Worked examples of the Secant-Tangent Theorem in advanced geometry

📚 Quick Study Guide

🔍 The Secant-Tangent Theorem relates the lengths of line segments created when a secant and a tangent intersect outside a circle. 💡 The theorem states: If a tangent segment and a secant segment are drawn to a circle from an external point, then the square of the length of the tangent segment is equal to the product of the length of the secant segment and its external segment. 📝 Mathematically: If $PT$ is a tangent and $PBA$ is a secant to the circle from an external point $P$, then $(PT)^2 = PA \cdot PB$. 📐 Remember that $PA$ is the external part of the secant, and $PB$ is the whole secant. ➕ This theorem is a crucial part of understanding circle geometry and is frequently used to solve problems related to lengths of segments.

✏️ Practice Quiz

1. What does the Secant-Tangent Theorem primarily relate?
1. Lengths of chords inside a circle
2. Angles formed by two tangents
3. Lengths of segments created by a secant and tangent from an external point
4. Areas of sectors
2. If $PT$ is a tangent to a circle from an external point $P$, and $PBA$ is a secant from the same point such that $PA = 4$ and $PB = 9$, what is the length of $PT$?
1. 3
2. 6
3. 8
4. 13
3. In the Secant-Tangent Theorem, which segment's square is equal to the product of the whole secant and its external part?
1. Chord segment
2. Tangent segment
3. Diameter segment
4. Radius segment
4. If the tangent segment $PT = 8$ and the whole secant $PB = 16$, what is the length of the external part of the secant, $PA$?
1. 2
2. 4
3. 6
4. 8
5. Given that $PT$ is tangent to circle $O$ at $T$, and secant $PBA$ intersects circle $O$ at $A$ and $B$. If $PA = 5$ and $AB = 7$, find the length of $PT$.
1. $\sqrt{35}$
2. 6
3. $\sqrt{60}$
4. $\sqrt{12}$
6. $PT$ is tangent to a circle at point $T$. Secant $PBA$ intersects the circle at points $A$ and $B$. If $PT = 10$ and $PA = 4$, find the length of $AB$.
1. 16
2. 21
3. 25
4. 30
7. From a point $P$ outside a circle, tangent $PT$ and secant $PAB$ are drawn. If $PT=12$ and $PB=18$, what is the length of the segment $AB$?
1. 6
2. 8
3. 10
4. 12