# 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