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Intersecting Secants Theorem

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Intersecting Secants Theorem

Study Guide

Key Definition

If two secant segments are drawn to a circle from an exterior point M, let MN be the external part and MO the entire secant (MO = MN + NO), and similarly let MP be the external part and MQ the entire secant (MQ = MP + PQ). Then the products of each external segment and its entire secant are equal: MN × MO = MP × MQ.

Important Notes

The theorem applies only to secant segments from the same exterior point.
Use the formula $MN \cdot MO = MP \cdot MQ$ to find unknown lengths.
Secants intersect the circle at two points, chords intersect inside.
External segment is the part of the secant outside the circle.
Always double-check your arithmetic calculations.

Mathematical Notation

$\cdot$ represents multiplication

$=$ signifies equality

$\angle$ represents an angle

$\triangle$ represents a triangle

Remember to use proper notation when solving problems

Why It Works

The theorem leverages the properties of similar triangles and proportionality due to the intersecting secants establishing equal products: $MN \cdot MO = MP \cdot MQ$

Remember

Use $MN \cdot MO = MP \cdot MQ$ to solve problems involving intersecting secants.

Quick Reference

Intersecting Secants Property:$MN \cdot MO = MP \cdot MQ$

Understanding Intersecting Secants Theorem

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

If two lines intersect a circle, forming secants, then $MN \cdot MO$ equals $MP \cdot MQ$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $MN = 10$, $NO = 17$, and $MP = 9$, find $PQ$.

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Two secant segments are drawn from point M to a circular field: one passes through N and O, and the other through P and Q. Here MN is the external segment and NO the internal part of the first secant; MP is the external segment and PQ the internal part of the second. If MN = 6 and NO = 8 (so MO = 14), and MP = 3, find the length PQ (the rope segment inside the field).

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Two secants from an exterior point M intersect a circle at N, O and at P, Q respectively. Given MN = 8 cm, NO = 2 cm (so MO = 10 cm), and MP = 5 cm, find the length PQ.

Challenge Quiz

Single Choice Quiz

Advanced

Determine $PQ$ given MN = 8 (external), NO = 15 (internal, so MO = 23), and MP = 10 (external).

Recap

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Review key concepts and takeaways