Question

in the circle with center O, the length of the chords are in ratio 5:3. if angle POQ=100°,find
I) angle ROS
II) ratio between the arc PQ and the arc RS
III) angle OQP​

Answer 1

Step-by-step explanation:

I m starting with III) I PART

angle OQP +angle OPQ+angle POQ = 180

(sum of internal angles of triangle)

here OP and OQ are equal so angle OQP and angle OPQ is same

and angle POQ is given

2*OQP=180-100

angle OQP=80/2=40 degrees.

now Part,

I)

LET THE ANGLE BETWEEN OR AND OS IS β

angle ROS=β,

now length of chord is equal to

$2r \times sin( \alpha \div 2)$

so PQ/RS = (2RSIN(α/2))/(2RSIN(β/2))

PQ/RS= SIN (5π/18)/SIN(β/2),

ratio of chords is given 5/3

5/3 = SIN (5π/18)/SIN(β/2),

SIN(β/2) = (3*SIN (5π/18))/5

=0.45962666587,

β/2= sin inverse 0.45962666587,

β/2= 27.36302°

β= 53 degree approximately

II)

formula for arc is (rα) where, r is radius and α is angle covered by radius going from one end to another

arc PQ/arc RS=R*(5π/9)/R*β

=5π/9β.

put the value of β from question no. one and your answer will be

5π/9(53π/180)

=(5*180)/9*53

= 100/53

so your answer will be approax 2.

!!HOPE IT HELPS!!

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