Question

AB and CD are two parallel chords of a circle which are on either sides of the centre.Such that AB=10cm and CD=24cm . Find the radius if the distance between AB and CD is 17cm.

Answer 1

see the image for the ans

Answer 2

• The radius of the given circle is equal to 13 cm .

Given :- AB and CD are two parallel chords of a circle which are on either sides of the centre, such that AB = 10cm and CD = 24cm . The distance between AB and CD is 17cm .

To Find :-

• The radius of the circle ?

Concept used :-

• The line segment from centre to the chord bisect the chord at 90° .
• According to pythagoras theorem in a right angled triangle :- (Perpendicular)² + (Base)² = (Hypotenuse)²

Solution :- (Refer to image for diagram.)

Let us assume that, radius of the circle is equal to r cm .

So,

→ OB = OD = r cm

Now,

→ AB = 10 cm

So,

→ AE = EB = 10/2 = 5 cm { The line segment from centre to the chord bisect the chord at 90° }

similarly,

→ CF = FD = 24/2 = 12 cm

Now, let OF is equal to x cm .

So,

→ OE = (17 - x) cm { since distance between both chord is equal to 17 cm }

Now, in right angled ∆OEB we have,

→ OB² = OE² + EB² { By pythagoras theorem }

putting values from above,

→ r² = (17 - x)² + 5²

using (a - b)² = a² + b² - 2ab in RHS,

→ r² = 289 + x² - 34x + 25

→ r² = x² - 34x + 289 + 25

→ r² = x² - 34x + 314 ----- Equation (1)

similarly, in right angled ∆OFD we have,

→ OD² = OF² + FD² { By pythagoras theorem }

putting values from above,

→ r² = x² + 12²

→ r² = x² + 144 ----- Equation (2)

comparing Equation (1) and Equation (2) now,

→ x² - 34x + 314 = x² + 144

→ 34x = 314 - 144

→ 34x = 170

→ 34x = 34 × 5

dividing both sides by 34,

→ x = 5 cm

therefore, putting value of x in Equation (2),

→ r² = x² + 144

→ r² = (5)² + 144

→ r² = 25 + 144

→ r² = 169

→ r² = (±13)²

square root both sides,

→ r = ± 13 cm

since length of radius of a circle can't be in negative . Therefore, we can conclude that, the radius of the circle is equal to 13 cm .

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