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# In the diagram below, BD = 9, \$CE = 7, [ABC] = 30, and [ADE] = 20. Find [ACD].

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In the diagram above, BD = 9, CE = 7, [ABC] = 30, and [ADE] = 20. Find [ACD].

Jan 27, 2020

#1
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The area of triangle ACD is 18.

Jan 27, 2020
#2
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Area( triangle(ABC) )  =  30

Area( triangle(ACD) )  =  x

Area( triangle(ABD) )  =  Area( triangle(ABC) )  +  Area( triangle(ACD) )

Area( triangle(ABD) )  =  30 + x

Area( triangle(ACE) )  =  Area( triangle(ADE) )  +  Area( triangle(ACD) )

Area( triangle(ACE) )  =  20 + x

Using the formaula that   Area  =  ½ · base · height    and calling the height  h,

--->     Area( triangle(ABD) )  =  ½ · 9 · h  =  30 + x

--->                                             h  =  (60 + 2x) / 9

--->     Area( triangle(ACE) )  =  ½ · 7 · h  =  20 + x

--->                                             h  =  (40 + 2x) / 7

Combining these equations:  (60 + 2x) / 9  =  (40 + 2x) / 7

Solving:                                     420 + 14x  =  360 + 18x

60  =  4x

x  =  15

The area of triangle(AD)  =  15.

Jan 27, 2020