Circles And Lines

Product of segments of one chord equals product of segments of other chord, The relationship is AP × PB = CP × PD, The theorem applies when chords intersect inside the circle

The intersecting chords theorem states that when two chords intersect inside a circle, the product of the segments of each chord are equal. This applies to any chords, not just diameters, and the intersection point can be anywhere inside the circle, not necessarily the center.

16

Step 1: For point inside circle: product = r² - d². Step 2: r = 5 cm, d = 3 cm. Step 3: Product = 5² - 3² = 25 - 9 = 16. This is true for any chord through point P inside the circle.

Height of the arc above the chord, Length of the chord

To find the diameter using the chord method, we need the chord length and the height (sagitta) of the arc. These create a perpendicular chord situation where the diameter is divided by the chord, allowing us to use the product theorem.

4 cm

Step 1: Area = length × width = 20 sq cm. Step 2: If new width = 5 cm, then new length = 20 ÷ 5 = 4 cm. Step 3: The chord intersection method helps construct this geometrically without calculation. The products of segments equal the area.