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In the diagram below, we have BE = 7, AB = 5, and [CBDA]=15. What is [CDA]?

[asy] size(125);pair C = (0,0);pair D = (4,0);pair EE = (2.5,0);pair A = (2.5,4);pair B = (2.5,2.5);draw(A--C--D--cycle);draw(C--B--D--cycle);draw(A--EE, dashed);draw(rightanglemark(B,EE,D,10));dot(Label(

Guest Oct 3, 2017

Best Answer 

 #3
avatar+4711 
+2

Also! smiley

 

Let's call side CD, which is the base of triangle CDA,  " b "  .

Let's call the area of CDA  " a " .

 

(1/2)(b)(7 + 5)  =  a

(1/2)(b)(12)  =  a

6b  =  a

b  =  a/6

 

(1/2)(b)(7)  =  a - 15

(7/2)(b)  =  a - 15

(7/2)b + 15  =  a

                                 Plug in  a/6  for  b .

(7/2)(a/6) + 15  =  a

(7/12)a + 15  =  a      Multiply through by  12 .

7a + 180  =  12a

180  =  5a

  36  =  a

hectictar  Oct 4, 2017
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4+0 Answers

 #1
avatar+234 
+1

Is CPhill's ghost composing an answer right now? Because apparently, CPhill can compose answers while not being online... LOL

Mathhemathh  Oct 3, 2017
 #2
avatar+76908 
+1

Parallel to CD, draw line EF through B

 

We have four triangles EBA, FBA, EBC and FBD....and their combined area is 15

 

And because triangle EAF  is similar to triangle CAD.....then EB = 5/12 CE  and BF = 5/12 ED ....and the height of EBA and FBA = BA = 5, while the height of EBC and FBD = EB = 7

 

So we have

 

Area EBA          +  Area of FBA   +       Area of EBC      +  Area of FBD  =    15

 

(1/2)(5/12)CE*BA + (1/2)(5/12)ED *BA  + (1/2)(5/12)CE *EB  + (1/2)(5/12)ED *EB  =  15

 

(1/2)(5/12)CE*5 + (1/2)(5/12)ED *5  + (1/2)(5/12)CE *7  + (1/2)(5/12)ED *7  =  15

 

[ (1/2)(5/12) CE * (5 + 7) ]  + [ (1/2)(5/12) ED * (5 + 7) ]  = 15

 

[ (1/2)(5/12) CE * (12) ]  + [ (1/2)(5/12) ED * (12) ]  = 15

 

[ (1/2)(5/12)* 12  [ CE + ED]  =  15

 

(60/24) [ CD]  = 15

 

CD  = 15 * (24/60) =   (1/4) * 24  =  6

 

So.....area CDA =  (1/2)CD * EA =   (1/2) * 6 * 12  =  36 units^2

 

 

 

cool cool cool

CPhill  Oct 4, 2017
 #3
avatar+4711 
+2
Best Answer

Also! smiley

 

Let's call side CD, which is the base of triangle CDA,  " b "  .

Let's call the area of CDA  " a " .

 

(1/2)(b)(7 + 5)  =  a

(1/2)(b)(12)  =  a

6b  =  a

b  =  a/6

 

(1/2)(b)(7)  =  a - 15

(7/2)(b)  =  a - 15

(7/2)b + 15  =  a

                                 Plug in  a/6  for  b .

(7/2)(a/6) + 15  =  a

(7/12)a + 15  =  a      Multiply through by  12 .

7a + 180  =  12a

180  =  5a

  36  =  a

hectictar  Oct 4, 2017
 #4
avatar+76908 
+1

Hectitar's answer was more concise.....I told my ghost not to be so long-winded!!!!!

 

 

cool cool cool

CPhill  Oct 4, 2017

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