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# Help?

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In rectangle$$WXYZ, A$$ is on side$$\overline{WX}$$ $$such that AX = 4, B is on side \overline{YZ} such that BY = 18, and C is on side \overline{XY} such that angle ACB= 90 and CY = 2CX. Find AB.$$

Help?

Noori

May 30, 2020

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Is there an image? Also is it possible if you can type that out of Latex as some of the information is not there. Very hard to read xd.

May 30, 2020
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geno1314 the answer was not correct any thing you missed?

Guest Jun 3, 2020
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AX = 4

BY = 18

Call  XC = x

then YC = 2x

Triangle(AXC)  is a right triangle   --->   AC2  =  AX2 + XC2  --->   AC2  =  42 + x2   --->   AC  =  sqrt(16 + x2)

Triangle(BYC)  is a right triangle   --->   BC2  =  BY2 + YC2  --->   BC2  =  182 + (2x)2   --->   BC  =  sqrt(324 + 4x2)

Area( trapezoid(AXYB) )  =  ½·(x + 2x)(4 + 18)  =  33x

Area( triangle(AXC) )  =  ½·4·x  =  2x

Area( triangle(BYC) ) =  ½·18·2x  =  18x

Area( triangle(ABC) )  =  Area( trapezoid(AXYB) )  -  Area( triangle(AXC) )  -  Area( triangle(BYC) )

=  33x - 2x - 18x  =  13x

Also:  area( triangle(ABC) )  ½·AC·BC  =  ½· sqrt(16 + x2)·sqrt(324 + 4x2)

=  ½· sqrt( (16 + x2) · (324 + 4x2) )

=  ½· sqrt( 4x4 + 388x2 + 5184 )

Therefore:     ½· sqrt( 4x4 + 388x2 + 5184 )  =  13x

sqrt( 4x4 + 388x2 + 5184 )  =  26x

4x4 + 388x2 + 5184  =  676x

4x4 - 288x2 + 5184  =  0

x4 - 72x2 + 1296  =  0

(x2 - 36)(x2 - 36  =  0

(x + 6)(x - 6)(x + 6)(x - 6)  =  0

--->   x  =  6

--->   XC  =  x   --->   XC  =  6

--->   YC  =  2x   --->   YC  =  12

--->   AC  =  sqrt(16 + x2)   --->   AC  =  sqrt(52)

--->   BC  =  sqrt(324 + 4x2)  --->   BC  =  sqrt(468)

--->  AB2  =  AC2 + BC2   --->   AB2  =  52 + 468  =  520   --->   AB  =  sqrt(520)

May 30, 2020