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How to Find the Length of the Chord? Common to Semicircle and Circle

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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 center of the semicircle, AM = 8 cm, CM = 2 cm and O is the center of the circle

Connect OM

AM × CM = OM²

⇒ 8 × 2 = OM²

⇒ OM² = 16

⇒ OM = 4 cm

So we get the radius of the circle = 4 cm

Also radius of semicircle = (8 + 2) / 2 = 5 cm

Now connect OP, OB, and PB, and then we get the triangle POB

From the figure

OP = 4 cm (radius of the circle)

OB = PB = 5 cm(radius of the semicircle)

Area of the 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 connects the center 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