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1) As shown in the diagram, NM is the median of trapezoid ABCD. We know AB = 5 and MN = 11. What is CD?

2) The figure shows a regular pentagon. Two sides are extended until they intersect as shown. What is the degree measure of angle ∠BPA?

3) A, B, C, and D are points on a circle, and segments AC and BD intersect at P, such that AP = 8, PC = 1, and BD = 6. Find BP, given that BP < DP.

 

4) In 4ABC, AB = 3 and BC = 5. Find the number of possible integer values of CA. [Disclaimer]: In this class, we do not consider degenerate triangles (i.e. a triangle whose area is 0).

 

5) In obtuse 4ABC, AB = 3 and BC = 5. Find the number of possible integer values of CA

 

Thanks!

 Mar 22, 2020
 #1
avatar+111438 
+2

1)  NM  is  the  midline  of the trapezoid

 

Its  length  is the  average  of the two base lengths....so....

 

(AB + DC)   / 2     = 11

 

AB + DC  =   22

 

5 + DC  = 22

 

DC   =  17

 

cool cool cool

 Mar 22, 2020
 #3
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Thanks!

Guest Mar 23, 2020
 #2
avatar+111438 
+2

(2)

 

The interior  angles of the pentagon each measure 108°

 

So....  base angles PBA  and PAB are both supplemental to 108°  = 72°

 

So  angle  BPA =   180 - 2(72)  =   180  -144   =  36°

 

 

cool cool cool

 Mar 22, 2020
 #4
avatar+111438 
+2

3) A, B, C, and D are points on a circle, and segments AC and BD intersect at P, such that AP = 8, PC = 1, and BD = 6. Find BP, given that BP < DP.

 

I believe  that we can use the intersecting chord theorem here.........we have that

 

AP * PC  = BP * DP

 

8 * 1  =   BP * DP

 

    8 =   BP * DP

 

We  know that   BD  = 6

 

So.....let   BP = x  and DP  =  6 - x

 

So

 

8 =  x * (6 - x)      simplify

 

6 = 6x  - x^2      rearrange as

 

x^2  - 6x  +  8   = 0      factor

 

(x - 4) ( x - 2)  = 0

 

Setting each factor to 0  and  solving for  x we get that  x  =  4 or  x  = 2

 

But BP < DP....so.....x =  2  must be correct ....and this equals  BP

 

 

 

 

cool cool cool

 Mar 23, 2020

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