My Math Forum Length of sides and angle calculations

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February 1st, 2010, 04:27 AM   #1
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Length of sides and angle calculations

Hi,

Can someone tell me how I would find a) the length of side BC and

b) the angle B,on the sketch,

Thanks....[attachment=0:3nsc7msn]01-02-2010 12-45-26_0063.jpg[/attachment:3nsc7msn]
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 01-02-2010 12-45-26_0063.jpg (398.2 KB, 135 views)

 February 1st, 2010, 11:38 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,485 Thanks: 2041 Is AC perpendicular to BD? You need some extra piece of information such as that.
 February 1st, 2010, 01:18 PM #3 Math Team   Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408 Re: Length of sides and angle calculations Hello, manich44! I explained this at another site. We are given three sides and one angle of a quadrilateral. [color=beige]. . [/color]This does not determine a unique quadrilateral. We find that:[color=beige] .[/color]$BD \:\approx\:60.5$ Then we have $\Delta BCD$, but we know only two of its sides. If $C \,=\,90^o$, we have a chance . . . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Hello, skipjack! I thought of that, too. But if $AC\,\perp\,BD$, we'd have a kite and $AD \,=\,CD.$
 February 2nd, 2010, 01:23 AM #4 Member   Joined: May 2009 Posts: 90 Thanks: 0 Re: Length of sides and angle calculations Sorry Guys, Forgot to include total area = 1276m2
 February 2nd, 2010, 07:19 AM #5 Global Moderator   Joined: Dec 2006 Posts: 20,485 Thanks: 2041 Area triangle ADB = (1/2)(34.37)(41.45)sin(105.24°)m² = 687.26831m² Area triangle CDB = 1276m² - 687.26831m² = 588.73169m² By the cosine rule, BD² = (34.37² + 41.45² - 2(34.37)(41.45)cos(105.24°))m² = 3648.3673m², so BD = 60.4017164m Area triangle CDB = (1/2)(BD)(CD)sin(angle BDC), so sin(angle BDC) = 2(588.73169)/((60.4017164)(37.62)) = 0.51817844, and so angle BDC = 31.210144° or 148.789856°. Now you can use the cosine rule to find the two possible values for the length of BC. You can then use the sine rule to find angles ABD and CBD, and hence angle ABC.
 February 3rd, 2010, 02:43 AM #6 Member   Joined: May 2009 Posts: 90 Thanks: 0 Re: Length of sides and angle calculations Hi, Thanks for the help. I get length BC = 34.30 and angle B 74.6 deg. Can you explain 31.210144° or 148.789856°. though Thanks...
 February 4th, 2010, 03:07 AM #7 Global Moderator   Joined: Dec 2006 Posts: 20,485 Thanks: 2041 Nothing in the problem requires that angle BDC is acute. How did you calculate 74.6° for angle ABC?

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