# The lengths of nonparallel sides of a trapezoid are equal to 6 and 10. It is known that a circle can be inscribed in trapezoid. Find length of bases.The midline divides the trapezoid into two parts...

The lengths of nonparallel sides of a trapezoid are equal to 6 and 10. It is known that a circle can be inscribed in trapezoid. Find length of bases.

The midline divides the trapezoid into two parts whose areas are related as 5 : 11.

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Let the trapezoid be named ABCD with `bar(AB)||bar(CD)` .

(1) Since the trapezoid is convex, in order for a circle to be inscribable in the trapezoid we must have `AB+CD=AD+BC`

(2) The midline divides the trapezoid into two areas in the ratio of 5:11. Call the areas I and II:

Let the shorter base be x and the longer base be y. Then the lengthof the midline is `(x+y)/2` . Let the height of the trapezoid be h; the distance from the midline to either of the bases is `h/2` . (If three parallel lines cut a transversal into congruent segments, they cut all tranversals into congruent segments.)

Then we can find the areas of the sections created by the midline:

`A_I=1/2(x+(x+y)/2)(h/2)`

`A_(II)=1/2((x+y)/2+y)(h/2)`

Now `(A_I)/(A_(II))=(1/2(x+(x+y)/2)(h/2))/(1/2((x+y)/2+y)(h/2))`

`=(3x+y)/(x+3y)`

We know this ratio is 5:11 so

`(3x+y)/(x+3y)=5/11==>7x=y`

(3) We now have a trapezoid with bases x and 7x, and nonparallel sides 6 and 10. Since we can inscribe a circle in the trapezoid:

`x+7x=6+10==>8x=16==>x=2`

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**The lengths of the bases are 2 and 14**

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