Segment Lengths in Circles
Fully Explained w/ 10 Examples!

Understanding segment lengths in circles is a crucial skill in geometry.

You’re in good hands because that’s what today’s lesson is all about.

You’re going to learn how to quickly find the segment lengths (chords, tangents, & secants) for circles.

Let’s get started!

In the previous lesson you learned how to find the arcs given two intersecting secants or tangents.

But now it’s time to investigate the length of two intersecting secants or tangents.

Did you know that three separate theorems help us solve these problems?

It’s true

1. Intersecting Chords Theorem

If two chords or secants intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

As seen in the image below, chords AC and DB intersect inside the circle at point E. Therefore, the product of the lengths of chord AC equals the product of the lengths of chord DB.

2. Secant Secant Theorem

Now when two secant segments have a common endpoint outside a circle, the product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant and its external part.

As seen in the graphic below, secants GP and FP intersect outside the circle at point P. Therefore, we can solve for unknown lengths by multiplying the external part (PH) by the entire secant length (PG) and set it equal to the product of the external part (PE) and the entire secant length (PF) of the second secant.

3. Tangent Secant Theorem

Thirdly, if a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

The image below illustrates this theorem by showing how the Tangent Secant Theorem and the Secant Secant Theorem are the same: external part (EA) times the whole (EA) is equal to the external part (EC) times the whole (ED).

We could also use the geometric mean to find the length of the secant segment and the length of the tangent segment, as Math Bits Notebook accurately states.

In the video below, you’ll use these three theorems to solve for the length of chords, secants, and tangents of a circle.

Video – Lesson & Examples

46 min

• Introduction to Video: Lengths of Intersecting Secants
• 00:00:30 – Theorems for finding segment lengths in circles (Examples #1-4)
• Exclusive Content for Member’s Only

• 00:14:09 – Find the indicated segment length (Examples #5-8)
• 00:28:27 – Find the indicated segment length given secants and tangents (Examples #9-10)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions