C i r c l e s . . Circles..

Geometry Level 2

A B AB and C D CD are chords perpendicular to each other. The radius of the circle is 7 cm. Find the sum of the length of the arcs A Y D AYD and B X C BXC .

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A s s u m p t i o n s : Assumptions:

Use π \pi = 22 / 7 22/7 .


The answer is 22.

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2 solutions

Christian Daang
Jan 17, 2015

S o l u t i o n : Solution:

Let's say, E E is the intersection of A B AB and C D CD . We know that C E B \angle CEB = 1 2 ( A r c C B + A r c A D ) \frac{1}{2}*(Arc CB + Arc AD)

--> A r c C B + A r c A D = 180 d e g r e e s Arc CB + Arc AD = 180 degrees .

This means, the length of A r c C B + A r c A D Arc CB + Arc AD is simply h a l f half of the circumference. C = ( 7 ) ( 2 ) ( π ) = 14 π C = (7)(2)(\pi) = 14\pi .

Therefore, A r c C B + A r c A D = C / 2 = 7 π u n i t s > 7 22 7 > 22 u n i t s \boxed{Arc CB + Arc AD = C/2 = 7\pi units -> 7*\frac{22}{7} -> 22 units}

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N o t e : Note:

A whole c i r c u m f e r e n c e circumference varies directly as the w h o l e whole sum of the angles in the circle or in short,

C = 360 k C = 360k

In our situation, C = 14 π > 14 π = 360 k C = 14\pi --> 14\pi = 360k

In the other hand, C = 180 k C = 180k

by using ratio and proportion, we obtain C = 7 π C = 7\pi or in conclusion, 7 π > 7 22 7 > 22 u n i t s \boxed{7\pi} -> 7*\frac{22}{7} -> 22 units is the s u m sum of the lengths of Arc CB & Arc AY.

Yash Singh
Jan 28, 2015

we can also do this by visualising , first let the chords perpendicular be the diameters the the length of the sum of arc of the two opposite quadrants formed will be half of the full circle and if we move the point of intersection away from the centre or vice versa , we will se that the length of the opposite arcs formed changes proportionately , one decreases and one increases but the sum remains a constant .

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