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November 8th, 2012, 05:03 PM  #1  
Newbie Joined: Nov 2012 Posts: 4 Thanks: 0  Calculation of sides and area of rightangled trapezoid
I have to calculate sides and area of rightangled trapezoid if I know only diagonals. The task is: Quote:
(Sorry for my English, I am not from Englishspeaking country.)  
November 9th, 2012, 01:23 AM  #2 
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Re: Calculation of sides and area of rightangled trapezoid
Here is a sketch Triangles ABS, BCS and CDS are similar since angle BAS = angle CBS = angle DCS (see colored dots) and all these triangles are rightangled. So we have DS/CS = CS/BS = BS/AS (1), BS + DS = BD = 9 and AS + CS = AC = 12 Shifting AC a length of CD along CD gives us (CD + AB)^2 = 9^2 + 12^2 = 225. By Pythagoras: CS^2 + DS^2 = CD^2 CS^2 + BS^2 = BC^2 BS^2 + AS^2 = AB^2 AS^2 + DS^2 = AD^2 Now, let DS = a and CS/DS = b. Then CS = ab, BS = ab^2 and AS = ab^3. Gives us BS + DS = ab^2 + a = 9 = a(b^2 + 1) and AS + CS = ab^3 + ab = b(ab^2 + a) = 9b = 12 i.e. b = 4/3, so a = 9/(b^2 + 1) = 9/((4/3)^2 + 1) = 9/(5/3)^2 = 81/25. Can you use this to find the sides? 
November 9th, 2012, 06:56 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 21,028 Thanks: 2259 
I've corrected some slips in the above post.

November 10th, 2012, 04:26 AM  #4  
Newbie Joined: Nov 2012 Posts: 4 Thanks: 0  Re: Calculation of sides and area of rightangled trapezoid
Hello, thanks very much for your response. But I don't understand this: Quote:
 I was thinking about my solution, here it is: Let AB = a, BC = b, CD = c and DA = d. Then Code: d = ?((ac)²  b²) Code: a² + b² = 12² = 144 b² + c² = 9² = 81 Code: b² = 144  a² = 81  c² b² = a²  c² = 63 Code: a² + b² = 144 a² + 63 = 144 a² = 81 a = 9 Code: b² + c² = 81 63 + c² = 81 c² = 18 Next thing, if AS = x, CS = 12x, DS = y and BS = 9y then due to Pythagoras Code: a² = 12x b² = 9*(9y) = 12*(12x) c² = 9y Code: 12x = y(9y) 9y = x(12x) 12x  9y = x²  y² Code: c²  y² = a²  x² x²  y² = a² c² ... b² = a² c² Thanks very much.  
November 10th, 2012, 12:42 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 21,028 Thanks: 2259 
There's more than one error. If you try to justify everything you have done, you should be able to spot the errors.

November 10th, 2012, 12:50 PM  #6  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  Re: Calculation of sides and area of rightangled trapezoid Quote:
 
November 10th, 2012, 01:46 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 21,028 Thanks: 2259 
Both expressions are valid.

November 10th, 2012, 05:46 PM  #8  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  Re: Quote:
 
November 11th, 2012, 03:00 AM  #9  
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  Re: Quote:
Here is what I used: Earlier, we found: DS/CS = CS/BS = BS/AS which gives CS/DS = BS/CS = AS/BS I set DS = a and CS/DS = b, yields, CS = CS * 1 = CS * (DS/DS) = DS * (CS/DS) = a * b = ab. (you don't need the bold brackets..) Likewise, BS = CS * (BS/CS) = ab * b = ab^2 and AS = BS * (AS/BS) = ab^2 * b = ab^3 Get it? Quote:
 
November 11th, 2012, 08:29 AM  #10  
Global Moderator Joined: Dec 2006 Posts: 21,028 Thanks: 2259  Quote:
 

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area, calculation, rightangled, sides, trapezoid 
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