Given a circle of radius ‘r’ with centre ‘O’. Chord ABmakes 90° angle at centre O. (Radius, r=\frac{5\sqrt2}{2}cm) Analyse the following statements:

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Solution

Q.
Given a circle of radius ‘ r ’ with centre ‘O’. Chord AB makes 90° angle at centre O .

$(R a d i u s, r = \frac{5 \sqrt{2}}{2} c m) A n a l y s e t h e f o l l o w i n g s t a t e m e n t s :$

(i) Difference between areas of two segments ADB and ACB made by chord AB is $\frac{25}{4} (π + 2) c m 2$

$(i i) A r e a o f Δ A O B = \frac{25}{2} {cm}^{2}$

$(i i i) L e n g t h o f c h o r d A B = 5 cm$

$(iv) Length of arc ACB = \frac{3 \sqrt{2} π}{4} cm$

Which of the above statements are correct Choose the correct option from the following:

1 (ii) and (iii)

2 (i) and (iv)

3 (iii) and (iv)

4 (i) and (iii)

Ans.
(4)

$Δ A O B i s r i g h t - a n g l e d a t O L e n g t h o f c h o r d A B : A B = \sqrt{A O^{2} + B O^{2}} = \sqrt{r^{2} + r^{2}} = \sqrt{2} r = \sqrt{2} \times \frac{5}{\sqrt{2}} = 5 cm (iii) is correct . Area o f Δ AOB : Area = \frac{1}{2} \times AO \times BO = \frac{1}{2} r^{2} = \frac{1}{2} {(\frac{5}{\sqrt{2}})}^{2} = \frac{25}{4} {cm}^{2} Thus (ii) is incorrect (because statement (ii) says \frac{25}{2}, not \frac{25}{4}) . Length of arc ACB : Arc length = \frac{θ}{360 °} \times circumference = \frac{90 °}{360 °} \times 2 πr = \frac{1}{4} \times 2 π \times \frac{5}{\sqrt{2}} = \frac{5 \sqrt{2} π}{4} cm But statement (iv) says \frac{3 \sqrt{2} π}{4}, so (iv) is incorrect . Finding areas of segments Let A_{1} = area of minor segment ACB Let A_{2} = area of major segment ADB Area of a segment : \frac{θ}{360 °} {πr}^{2} - \frac{1}{2} r^{2} Forθ = 90 ° : A_{1} = [\frac{π}{360} \times 90 - \frac{1}{2}] r^{2} = [\frac{π}{4} - \frac{1}{2}] {(\frac{5}{\sqrt{2}})}^{2} A_{1} = (\frac{25 π}{8} - \frac{25}{4}) {cm}^{2} Area of entire circle : {πr}^{2} = π \times {(\frac{5}{\sqrt{2}})}^{2} = \frac{25 π}{2} A_{2} = \frac{25 π}{2} - A_{1} A_{2} = \frac{25 π}{2} - (\frac{25 π}{8} - \frac{25}{4}) = \frac{75 π}{8} + \frac{25}{4} Required difference : A_{2} - A_{1} = (\frac{75 π}{8} + \frac{25}{4}) - (\frac{25 π}{8} - \frac{25}{4}) = \frac{25}{4} (π + 2) {cm}^{2} ∴ (i) is correct$

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