Intersecting Chords Study Notes

In 10th-grade geometry, understanding the relationships formed when chords intersect within a circle is crucial. This topic explores the properties of segments created by these intersections.

Intersecting Chords Theorem 📐

When two chords intersect inside a circle, the product of the lengths of the segments on each chord is equal. This is a fundamental theorem for solving problems involving intersecting chords.

Statement of the Theorem:

If two chords, say \( \overline{AC} \) and \( \overline{BD} \), intersect at a point \( P \) inside a circle, then the following relationship holds: \[ AP \cdot PC = BP \cdot PD \]

Explanation:

• \( \overline{AC} \) and \( \overline{BD} \) are chords of the circle.
• \( P \) is the point of intersection of these two chords within the circle.
• \( AP \) and \( PC \) are the two segments of chord \( \overline{AC} \).
• \( BP \) and \( PD \) are the two segments of chord \( \overline{BD} \).
• The theorem states that 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.

Visual Representation:

Imagine a circle with two lines crossing inside it. Each line is cut into two pieces by the crossing point. The theorem tells us how the lengths of these pieces relate to each other.

Example Problem 📝

Consider a circle where two chords, \( \overline{XY} \) and \( \overline{WZ} \), intersect at point \( Q \). If \( XQ = 4 \) cm, \( QY = 6 \) cm, and \( WQ = 3 \) cm, find the length of \( QZ \).

Solution:

Using the Intersecting Chords Theorem:

\[ XQ \cdot QY = WQ \cdot QZ \]

Substitute the given values:

\[ 4 \cdot 6 = 3 \cdot QZ \] \[ 24 = 3 \cdot QZ \]

Solve for \( QZ \):

\[ QZ = \frac{24}{3} \] \[ QZ = 8 \]

Therefore, the length of segment \( QZ \) is 8 cm.

Key Takeaways 📌

• The intersection point must be inside the circle for this theorem to apply.
• The theorem relates the lengths of the segments of the chords, not the lengths of the chords themselves.
• This theorem is a direct consequence of similar triangles formed by the intersecting chords and the arcs they subtend.

Practice Problems 💡

Solve the following problems to reinforce your understanding:

Problem	Given Information	What to Find
1	Chords \( \overline{AB} \) and \( \overline{CD} \) intersect at \( E \). \( AE = 5 \), \( EB = 10 \), \( CE = 2 \).	Length of \( ED \).
2	Chords \( \overline{PQ} \) and \( \overline{RS} \) intersect at \( T \). \( PT = 7 \), \( TQ = 4 \), \( RT = 8 \).	Length of \( TS \).
3	Chords \( \overline{MN} \) and \( \overline{OP} \) intersect at \( U \). \( MU = 6 \), \( UN = 9 \), \( OU = UP \).	Length of \( OU \) (and \( UP \)).