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1. In trapezoid ABCD with AB || CD, the diagonals intersect at point P. The line passing through P and parallel to AB intersects AD and BC at points E and F, respectively. Given that AB = 6 and CD = 12, find EP and FP. 

2. The lengths of two sides of an isosceles triangle are 10 and 12 centimeters. Find all possible values of the area of this triangle. 

3. The area of trapezoid ABCD, with AD||BC, is 164. The altitude is 8, AB is 10, and CD is 17. Find all possible values of the length of side BC, in centimeters. 

 Dec 7, 2020
 #1
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I really need the answers!

 Dec 7, 2020
 #2
avatar+114141 
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2. The lengths of two sides of an isosceles triangle are 10 and 12 centimeters. Find all possible values of the area of this triangle. 

 

Either  the remaining side is  10  or it is 12

 

If it is 10

 

Semi-perimeter, S  =   (10 + 10 + 12)  / 2  =   16

 

Area  = sqrt  [ S ( S - A)  (S - B)  (S - C)  ]    where A,B,C  are the side lengths

 

Area =   sqrt [  16  ( 16 - 10)  (16-10)(16-12)  ]  =   sqrt [ 16 * 6 * 6 * 4 ]  =  sqrt [ 16 * 36 *4 ] = 4*6*2  = 48

 

If it is 12

 

Semi-perimeter  =  (10 + 12 + 12)  / 2 =  17

 

Area  = sqrt [  17  (17 -12) (17-12) ( 17 -10)  ]  =  sqrt  [  17 * 5 *5 * 7  ]  =  sqrt [ 119 * 25 ]  =  5sqrt (119)

 

cool cool cool

 Dec 7, 2020
 #3
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1. By similar triangles, EP = 4/7 and FP = 5/3.

 Dec 8, 2020
 #4
avatar+114141 
+1

EP  and FP  could have the  same  value  (in this case  EP  = FP  = 4)

 

See here  :

 

 

This problem  would  have a definite answer if  AD  or BC    lengths were  specified

 

cool cool cool

 Dec 8, 2020

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