Question

If the arcs if the same length in 2 circles subtend angles 65° and 70° at the centre, find the ratio of their radii
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Answer 1

Answer:

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Step-by-step explanation:

$so \: here \: for \: a \: certain \: circle \: \\ its \: two \: arcs \: are \: of \: equal \: length \: \\ so \: let \: those \: two \: arcs \: be \: l1 \: and \: l2 \\ hence \: \: l1 = l2 \: \: \: \: \: \: \: \: \: \: \: \: (1)$

$so \: for \: l1 \: its \: radius \: be \: r1 \: and \: for \: l2 \: be \: r2 \\ so \: here \: let \: central \: angle \: of \: l1 \: be \: \: θ1 \: and \: that \: of \: other \: be \: θ2 \\ so \: here \: θ1 = 65 \\ θ2 = 70$

$so \: we \: know \: that \: \\ length \: of \: an \: arc = θ/360 \times 2\pi.r \\ hence \: as \: \: l1 = l2 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: from \: (1) \\ θ1/360 \times 2\pi.r1 = θ2/360 = 2\pi.r2 \\ θ1.r1 = θ2.r2 \\ 65.r1 = 70.r2 \\ ie \: \: r1/r2 = 70/65$

$thus \: \: r1/r2 = 14/13 \\ hence \: ratio \: of \: these \: radii \: of \: arcs \: of \: equal \: lengths \: is \: 14 : 13$

Answer 2

$r_1 : r_2 = 14 : 13$

Step-by-step explanation:

Let arc lengths radii and angles of the two crcles be ,

$l_1, l_2, \: \: r_1, r_2 \: \: and \: \theta_1, \theta_2 \: \: respectively.$

Then,

$l_1 = r_1 \times \theta_1 \: \: \: and \: \: l_2 = r_2 \times \theta_2$

Here, Arc lengths are equal.

Therefore,

$r_1 \times \theta_1 \: \: = r_2 \times \theta_2$

$\frac{r_1}{r_2} = \frac{ \theta_2}{ \theta_1}$

Therefore,

$r_1 : r_2 = 70 : 65 \: \\ \implies r_1 : r_2 = 14 : 13.$

Hence, the answer.