April 9th, 2018, 08:33 AM  #1 
Senior Member Joined: Aug 2014 From: India Posts: 295 Thanks: 1  How ratios are written here?
In the below figure, O is the center of the circle. Radius is 7 cm. $\angle$ BOC = 120 Find the length of BC? Solution: Join AO to meet BC in M $\triangle ABC \cong \triangle AOC$ $\triangle BMO \cong \triangle CMO$ $\angle BMO$ = 90 Triangle BMO has three angles are 90, 60 and 30. The sides are in the ratio of 2: $\sqrt 3$ : 1 2X = 7 $\Rightarrow$ X = 7/2 BM = $\Rightarrow$ $\sqrt 3$ x 7 / 2 = 7$\sqrt 3 / 2$ BC = 2 x 7$\sqrt 3 / 2$ = 7 $\sqrt 3$ My Question is: How ratios are written here? Last edited by Ganesh Ujwal; April 9th, 2018 at 08:36 AM. 
April 9th, 2018, 12:20 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,761 Thanks: 1416 
Latex input \$\dfrac{a}{b}\$ will yield $\dfrac{a}{b}$

April 9th, 2018, 06:36 PM  #4 
Senior Member Joined: Aug 2014 From: India Posts: 295 Thanks: 1  
April 9th, 2018, 06:43 PM  #5  
Senior Member Joined: Aug 2014 From: India Posts: 295 Thanks: 1  Quote:
Don't tell me I told $\angle$ BMO is 90. Use your own method and prove me.  
April 9th, 2018, 11:27 PM  #6 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,852 Thanks: 750 Math Focus: Wibbly wobbly timeywimey stuff.  
April 10th, 2018, 01:33 AM  #7 
Senior Member Joined: Aug 2014 From: India Posts: 295 Thanks: 1 
How is the ratio obtained at the step shown in the solution?

April 10th, 2018, 03:04 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,291 Thanks: 1683 
The diagram is symmetrical about AO, so angles BMO and CMO have equal measure, which implies they are right angles; also angles BOM and COM have equal measure, which implies they are both 60$^\circ$. Knowing the angles of triangle BMO are 90$^\circ$, 60$^\circ$ and 30$^\circ$ implies that it is half of an equilateral triangle. 
April 10th, 2018, 07:33 AM  #9 
Senior Member Joined: Aug 2014 From: India Posts: 295 Thanks: 1  My question is different. How ratios are obtained? I got the answer myself: http://mathcentral.uregina.ca/qq/dat....05/gary1.html Can you tell me how 2X = 7 is obtained in my solution? Last edited by Ganesh Ujwal; April 10th, 2018 at 07:46 AM. 
April 10th, 2018, 09:17 AM  #10 
Senior Member Joined: Aug 2014 From: India Posts: 295 Thanks: 1 
I got the answer myself: 2x = 7 because 2 x is hypotenuse. Here hypotenuse is radius of the circle i.e 7.


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