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avatar+1681 

\(In\ the\ diagram,\ angle\ A\ is\ right,\ E\ is \ the\ midpoint\ of\ AB,\ D\ is\ the\ midpoint \\ of\ AC,\ AB = 16\ and\ AC = 12. \\ \\What\ is\ the\ area \ of\ AEFGD,\ where \ EF\ and\ DG\ are\ perpendicular\ to\ BC?\)

 

Image:

 

I've labeled everything on paper then I have no idea what to do...

I solved for CB which is 20, but I can't get past that...

I would love it if you could help me.

 Aug 13, 2019
 #1
avatar+19793 
0

Calculate the area of the LARGE triangle...then subtrat the two smaller triangles.

 

You can calculate the angles of ADE.....find the hypotenuse....you are given two sides

    then you can calculate the angles FEB  CDG   etc

       Example : 180 - AED - 90  = FEB       to find the side lengths of those triangles and their areas......etc,,,

 

 

Or calculate the area AED and add the rectangle when you find the side lengths.....

 Aug 13, 2019
edited by ElectricPavlov  Aug 13, 2019
 #2
avatar+1681 
0

how do you find the side lengths?

tommarvoloriddle  Aug 13, 2019
 #3
avatar+28288 
+3

Find the lengths as follows:

 

 Aug 13, 2019
 #4
avatar+1681 
+1

So how do you find CG?

I'm stuck there.

ANd Fb...

tommarvoloriddle  Aug 14, 2019
 #5
avatar+28288 
+3

From (1)  t = 10

 

So from (2)  x + y = 10    ...(5)

 

Subtract (3) from (4)    y2 - x2 = 28  or  (y+x)(y-x) = 28   ...(6)

 

Use (5) in (6) to get    y - x = 28/10    or  y - x = 14/5   ...(7)

 

Add (5) and (7) to get    2y = 64/5  or   y = 32/5 

 

Subtract (7) from (5) to get    2x = 36/5   or  x = 18/5

 

Can you take it from there?

Alan  Aug 14, 2019
 #6
avatar+1681 
+2

What are the numbers?

like the (1)

tommarvoloriddle  Aug 14, 2019
 #7
avatar+28288 
+1

They label the equations.  So (1) means the equation labelled ...(1).

Alan  Aug 14, 2019
 #9
avatar+1681 
+1

Wait- you said 36/5=2x then you went like x=18/5... ?????

tommarvoloriddle  Aug 14, 2019
 #10
avatar+28288 
+3

If you have 2x = 36/5 you can divide both sides by 2 to get x on its own.  Dividing 36/5 by 2 gives 18/5, because 2 goes into 36 18 times.

Alan  Aug 14, 2019
 #11
avatar+1681 
0

oh whoopps i thought is was simplifing...

tommarvoloriddle  Aug 14, 2019
 #12
avatar+1681 
0

ok I still don't get it... :(

tommarvoloriddle  Aug 14, 2019
 #13
avatar+28288 
+2

John McEnroe's famous expression comes to mind!   

 

Are you seriously saying that you don't know how to divide a fraction by 2?

Alan  Aug 14, 2019
 #16
avatar+1681 
0

umm I do know but like the whole thingy

tommarvoloriddle  Aug 15, 2019
 #8
avatar+1681 
0

:o let me label my doagram first

 Aug 14, 2019
 #14
avatar+105464 
+2

Note that  triangle CAB  is a 12 - 16 - 20   Pythagorean Right triangle  where BC  = 20

 

Triangles  CGD  and EFB  are similar  to triangle CAB

 

Since CD  = 1/2 AC....then  CD = 6

So

DG/ CD  = AB/ CB

DG / 6 =  16/ 20

DG = 6* 16/20 =  6 * 4/5  = 24/5   = 4.8

 

And CG = sqrt (CD^2 - DG^2)  = sqrt  ( 6^2 - 4.8^2)   = sqrt (12.96) = 3.6

 

Likewise

Since EB  =  1/2 AB....then   EB  = 8

So

FB/ EB  = AB/CB

FB/ 8 =  16/20

FB = 8*16/20  = 8*4/5  = 32/5  = 6.4

So

GF = BC - GC - FB   = 20  - 3.6 - 6.4  = 10

 

So  [CFDE]  = DG * GF  = 4.8 * 10  = 48

 

And  [ DAE ]  =(1/2)(1/2)(AC) (1/2)(AB) =  1/8 (12 )(16)  = 24

 

So

 

[ AEFGD]  = [CFDE] + [DAE]  = 48 + 24 =   72

 

 

cool cool cool

 Aug 15, 2019
 #15
avatar+1681 
+1

Oh ok I get it now thank you CPHILL

tommarvoloriddle  Aug 15, 2019

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