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# help!

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The rectangle in the diagram has an area of 640 cm^2.  Points B and F are the midpoints of sides AC and AE, respectively.  What is the area of triangle BDF? May 4, 2020

#1
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well we know that FED+ DCB count for 50% of the total area, since they each are 25%. Further, the ABF triangle is 1/8 of the total, so we have 5/8 unshaded. 640/8 = 80.

This is because if we put points...

Let's say point X is Midpoint ED, Y is midpoint CD

BCD is one half BCDX, and BCDX is one half whole rect, so 1/4

Same applies to FED.

If we put a point Z at midpoint of FY and BX, AFB is one half ABFZ, which is 1/4 whole rect, so 1/8.

3*80 = 240

if you don't understand anything feel free to ask.

May 4, 2020
edited by hugomimihu  May 4, 2020

#1
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well we know that FED+ DCB count for 50% of the total area, since they each are 25%. Further, the ABF triangle is 1/8 of the total, so we have 5/8 unshaded. 640/8 = 80.

This is because if we put points...

Let's say point X is Midpoint ED, Y is midpoint CD

BCD is one half BCDX, and BCDX is one half whole rect, so 1/4

Same applies to FED.

If we put a point Z at midpoint of FY and BX, AFB is one half ABFZ, which is 1/4 whole rect, so 1/8.

3*80 = 240

if you don't understand anything feel free to ask.

hugomimihu May 4, 2020
edited by hugomimihu  May 4, 2020
#3
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Very elegant, Hugo   !!!!!

WAY better than  my method    !!!!!   CPhill  May 4, 2020
#4
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Thank you!

But TBH, I think yours might be a bit better, but both work, and the other can be used as a different strategy to check:

essentially we did the same thing, you just subtracted the unshaded, I calculated proportion of unshaded and subtracted that from 1.

hugomimihu  May 4, 2020
#2
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Area  of rectangle =   2(AB) * 2(AF)

Area of triangle ABF  =(1/2)(AB)(AF) =  (1/2)(AF)(AB)

Area of triangle BCD = (1/2) *2((AF) (AB)  = (AF)(AB)

Area of triangle FED  = (1/2) * 2(AB)(AF)  = (AF)(AB)

So   area  of  rectangle  = 2(AF) 2(AB)  = 4(AF)(AB)

So

640  = 4 (AF)(AB)   divide through by 4

160  =(AF)(AB)

So

The  area  of the  three right triangles =  (AF)(AB)  ( 1/2+ 1 + 1)  =  (5/2)(AF)(AB)  = (5/2)(160)  = 400

So....the area  triangle BDF    =  640  -  400  =   240   May 4, 2020