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Intersecting Secant-Tangent Theorem

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Intersecting Secant-Tangent Theorem

Study Guide

Key Definition

If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment: $\overline{UV}^2 = \overline{UX} \cdot \overline{UY}$

Important Notes

The tangent segment is a line that touches the circle at exactly one point: $\overline{UV}$
The secant segment passes through the circle: $\overline{UY}$
The external part of the secant segment is the portion outside the circle: $\overline{UX}$
Remember: $\overline{UV}^2 = \overline{UX} \cdot \overline{UY}$
This theorem only applies when both a tangent and secant are from the same exterior point

Mathematical Notation

$\overline{AB}$ represents a line segment

$+$ represents addition

$\cdot$ represents multiplication

$x^2$ represents x squared

$√(x)$ represents the square root of x

Remember to use proper notation when solving problems

Why It Works

By constructing the radius to the point of tangency and examining the resulting right angles, one can establish similar triangles between the tangent and secant segments. Alternatively, the power-of-a-point theorem directly yields $\overline{UV}^2 = \overline{UX}\cdot\overline{UY}$, showing the relationship holds for any circle. This fundamental result connects angle and segment properties in circle geometry.

Remember

Always use $\overline{UV}^2 = \overline{UX} \cdot \overline{UY}$ when solving problems involving a tangent and a secant from an exterior point.

Quick Reference

Tangent-Secant Theorem:$\overline{UV}^2 = \overline{UX} \cdot \overline{UY}$

Understanding Intersecting Secant-Tangent Theorem

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

When a tangent and secant are drawn from the same external point, the square of the tangent segment equals the product of the external secant part and the entire secant. For example, if $UX = 3$ and $XY = 5$, then $UV^2 = 3 \times (3+5) = 24$, so $UV = 2\sqrt{6}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $\overline{UX} = 8$ and $\overline{XY} = 10$, what is $\overline{UV}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are standing outside a circular pond and you see a straight path (tangent) and a bridge (secant) leading from your standing point to the pond.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a circle with a tangent and secant line drawn from the same external point. How can you prove the Intersecting Secant-Tangent Theorem?

Challenge Quiz

Single Choice Quiz

Advanced

Given $\overline{UX} = 7$, $\overline{XY} = 24$, find $\overline{UV}$.

Recap

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Review key concepts and takeaways