How do you apply the intersecting chords theorem?

To apply the intersecting chords theorem, multiply the segments of one chord and set it equal to the product of the segments of the other chord.

The intersecting chords theorem states that if two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal. Imagine you have a circle with two chords, AB and CD, intersecting at point E. The theorem tells us that the length of AE multiplied by the length of EB is equal to the length of CE multiplied by the length of ED. Mathematically, this is written as \( AE \times EB = CE \times ED \).

To use this theorem, first identify the point where the chords intersect. Measure or note the lengths of the segments created by this intersection. For example, if chord AB is divided into segments AE and EB, and chord CD is divided into segments CE and ED, you would measure these lengths. Then, simply multiply the lengths of the segments of one chord and set this product equal to the product of the segments of the other chord.

For instance, if AE = 3 cm, EB = 4 cm, CE = 2 cm, and ED = x cm, you would set up the equation \( 3 \times 4 = 2 \times x \). Solving this, you get \( 12 = 2x \), so \( x = 6 \) cm. This means the segment ED is 6 cm long. This method is useful for finding unknown lengths in problems involving intersecting chords within a circle.