maths-proof--10 points

2007-05-06 1:49 pm

1. PM is a tangent segment to a circle with centre O. Segment OP intersects the circle at N. If MO=MN, prove that N bisects OP.

2.Quadrilateral ABCD is cyclic(pic:http://hk.geocities.com/sinyeeangela/DSC00060_1.jpg), with perpendicular diagonals AC and BD intersecting at E. Point M is midpoint of CD. Prove that the line through M and E is perpendicular to AB.

3.The diagram contains three squares. Prove that x+y=z
(pic:http://hk.geocities.com/sinyeeangela/25.jpg)

更新1:

第三題有無其他方法？

回答 (1)

Tommy T.K. Lau

2007-05-06 5:45 pm

✔ 最佳答案

1.

圖片參考：http://i169.photobucket.com/albums/u209/lautszki/tangent1.jpg

Let OM = r.
ON = OM = r (radii)
MN = OM = r (given)
In △POM,
∠POM = 60° (prop. of equil. △)
∠OMP = 90°(tangent⊥radius)
cos 60° = r ÷ OP
OP = r ÷ cos 60° = 2r
NP = OP – ON = 2r – r = r
∴ ON = NP,
i.e. N bisects OP.

3.
Let a side of a square be h.
tan z = h / h = 1
z = 45°
tan y = h / 2h = 1/2
y = tan-1(1/2)
tan x = h / 3h = 1/3
x = tan-1(1/3)
x + y = tan-1(1/2) + tan-1(1/3)
= 45°
∴ x + y = z

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