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# geometry

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A quadrilateral has two pairs of congruent sides and a longer diagonal of length 6, as shown. For what value of x will the area of the shaded area be 40% of the area of the unshaded region?

May 1, 2019

#1
+12528
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A quadrilateral has two pairs of congruent sides and a longer diagonal of length 6, as shown. For what value of x will the area of the shaded area be 40% of the area of the unshaded region?

May 2, 2019
#2
+128845
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We only need to look at the top  two triangles.....they are under the same height.....and the areas of triangles under the same height are to each other as their bases

So.......

x / (6 - x)  = .40      cross-multiply

x =  .40 (6 - x)

x = 2.4 - .4x

1.4x  = 2.4

x = 2.4/1.4  =   12/7

Proof :   [ 2*  area of triangle on the left ] /  [ 2 * area of triangle on the right ]  =

[ 2 * triangle height * base length of left triangle ]  / [ 2 * triangle height * base length of right triangle ] =

[ base length of triangle on the left ]  / [ base length of triangle on the right ]

[12/ 7 ] / [ 6 - 12/7 ]   =

[ 12/7 ] / [ 42/7 - 12/7 ]  =

[12/7 ] / [ 30/7]   =

[ 12/7] * [ 7/30 ]  =

12/30  =

4/10   =

.40  = 40%

May 2, 2019