How to Find the length of the chord? common to semicircle and circle
Find the length of the chord

Find the length of the chord from the figure, If PQ is the common chord of the semicircle and circle
AC is the centre of the semicircle, AM = 8 cm, CM = 2 cm and O is the centre of the circle
Solution: length of the chord

Connect OM

Apply intersecting chords theorem in the figure, then
AM × CM = OM²
⇒ 8 × 2 = OM²
⇒ OM² = 16
⇒ OM = 4 cm
So we get the radius of circle = 4 cm
Also radius of semicircle = (8 + 2) / 2 = 5 cm
Now connect OP, OB and PB then we get the triangle POB

From the figure
OP = 4 cm (radius of the circle)
OB = PB = 5 cm(radius of the semicircle)
So we can find the area of the triangle POB using the heron’s formula
Area ofthe triangle using heron’s formula is √(s(s–a)(s–b)(s–c)
Here, s = (a + b + c) / 2, a = 5 cm, b = 5 cm and c = 4 cm
Then s = (5 + 5 + 4)/2 = 7
So area = √84 cm²
Connect PQ

PQ is the common chord and OB is connecting the centre of the circle and semicircle so OB is the perpendicular bisector of the chord PQ
so, PQ = 2 × PD
also, the Area of the triangle POB = ½ × OB × PD
That is, √84 = ½ × 5 × PD
So we get PD = 2√84 ÷ 5
Now, PQ = 2 × PD
That is, PQ = (8√21)/5 cm