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Quadrilateral ABCD is an isosceles trapezoid, with bases AB and CD. A circle is inscribed in the trapezoid, as shown below. (In other words, the circle is tangent to all the sides of the trapezoid.) The length of base AB is 2x, and the length of base CD is 2y. Prove that the radius of the inscribed circle is \(\sqrt{xy}.\)

 

 

Please provide an explanation with your answer! Thanks!

 Aug 1, 2020
 #1
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Hi there! 

After I read your question my intuall thoughts were I know this question. So i decided to help you by giving you i think everything you need to answer this yourself. So here is my points in order in which they should help you.

 

1. Because the circle is tuching both of the trapezoid bases what does this tell you about the raidious and the height?

2. Next find the sides of AD and BC

3. Use rule of tangents (google if you dont know)             

4.Draw a line from points A and B down to line DC. Use pytagorean theorem

5. REMBER that line that you drew in the previous step is also 2 times the radious. 

6. After you simplify the equation you get from these last 2 steps you prove your point. 

Notes: Rember AAS congruence and please try your best to word your answer in the right way make sure you prove why each step should happen. 

I understand were you are coming from so I gave you hints not an answer. I dont want to hand you an answer because of two reasions. 1. because you should figure it out on your own (with help because this is tricky) 2. Because i dont know how to word this without telling you what I wrote my self. 

Please take my hints and figure out your answer. 

Hope this helped you!!!

~Wolf :D

 Aug 1, 2020
 #2
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Quadrilateral ABCD is an isosceles trapezoid, with bases AB and CD. A circle is inscribed in the trapezoid, as shown below. (In other words, the circle is tangent to all the sides of the trapezoid.) The length of the base AB is 2x, and the length of the base CD is 2y. Prove that the radius of the inscribed circle is sqrt(xy).

 

r = sqrt( x*y )    or     r2 = x*y                   x < r < y

 

r = 2         x = 1.25          y = r2 / x       y = 3.2

 

Important:     AD = BC = x + y

 

Proof:        sqrt [( y-x )2 + ( 2r )2 ] = x + y

 

                                 4.45 = 4.45  laugh

 

This is one way to prove it.smiley

 Aug 2, 2020
edited by Dragan  Aug 2, 2020
edited by Dragan  Aug 2, 2020
 #3
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wow thats much easier than what I tried to explane. Nice job!

~Wolf :D

Guest Aug 2, 2020
 #4
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Thank you so much!

Otterstar  Aug 2, 2020

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