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!

Guest Mar 22, 2020

#1**+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

CPhill Mar 22, 2020

#2**+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°

CPhill Mar 22, 2020

#4**+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

CPhill Mar 23, 2020