Exploring Segments from Secants

Objective:

Students will be able to understand and apply the Two Secants Segments Theorem by correctly identifying and calculating proportional segments created by two secants intersecting outside a circle.

Assessment:

Students will demonstrate mastery of the objective by solving problems involving segment lengths created by two secants intersecting outside a circle.

Key Points:

• Two Secants Segments Theorem: When two secants intersect outside a circle, the circle divides the secants into segments that are proportional with each other.

• Proportional Segments: Segments created by two secants intersecting outside a circle are proportional.

• Formula: If two secants intersect outside a circle, the product of the lengths of one secant segment and its external segment is equal to the product of the other secant segment and its external segment.

Opening:

• Introduction to the topic with a real-world scenario involving secants and a circle.

• Question: How can we determine the lengths of segments created by secants intersecting outside a circle?

Introduction to New Material:

• Explaining the significance of the Two Secants Segments Theorem.

• Demonstrating how to identify and calculate proportional segments created by two secants intersecting outside a circle.

• Common Misconception: Thinking that secant segments are not proportional when intersecting outside the circle.

Guided Practice:

• Providing examples for students to work through in pairs.

• Guiding students from simple to complex problems.

• Monitoring student performance through questioning and observation.

Independent Practice:

• Assigning practice problems for students to work on individually.

• Students will be expected to calculate segment lengths using the Two Secants Segments Theorem.

• Monitoring student progress and providing support as needed.

Closing:

• Summarising the key points of the lesson through a quick group discussion.

• Reviewing the application of the Two Secants Segments Theorem.

Extension Activity:

• Create a set of challenging problems for students to explore further applications of the Two Secants Segments Theorem.

Homework:

• Homework suggestion: Assign a worksheet with various problems involving segments from secants for further practice.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG-C.A.4: Identify and describe relationships among inscribed angles, radii, and chords.

• CCSS.MATH.CONTENT.HSG-C.A.4: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Here is an example problem that demonstrates the application of the Two Secants Segments Theorem:

Example Problem:
In the diagram below, two secants, AB and CD, intersect outside a circle. If the length of segment AC is 8 cm, segment BC is 6 cm, and segment AD is 5 cm, what is the length of segment BD?

(Diagram not to scale)

        A---------B

        /           \

       /             \

      /               \

     /                 \

    /                   \

   O                     \

    \                   D

     \                 /

      \               /

       \             /

        \           /

         C---------/

Solution:
According to the Two Secants Segments Theorem, the product of the lengths of one secant segment and its external segment is equal to the product of the other secant segment and its external segment. Mathematically, this can be represented as:

AC * BC = AD * BD

Substitute the given values:
8 cm * 6 cm = 5 cm * BD
48 = 5 * BD
BD = 48 / 5
BD = 9.6 cm

Therefore, the length of segment BD is 9.6 cm.