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!
+128475 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
+128475 (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°
+128475 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
