急~~~中二MATHS~~~20分

兩條maths題

1). ABCD is a square and E is a point such that CDE is an equilateral triangle.

a).
Draw a diagram to represent the given information (two possible cases).

i can just tell u about the diagram

Case1:
E is a point inside the square

Case2:

E is a point outside the square.


b).
Is △ABE an isosceles triangle? Find suitable angles to verify your judgement.

Case1:

consider △AED and △BEC

AD = BC(prop of square)

ED = EC(sides of equilateral triangle)

angle EDA = angle ECB = 30

so,△AED and △BEC is a pair of equal triangle(A.S.S)

so,AE = EB(side corresponding to equal △)

Case2:

consider △BCE

BC = CE (given)

angle BCE

= 90 + 60

= 150

so,
angle EBC = 15(base angle at isos triangle)

angle of ABE

= 90 - 15

= 75

consider △ACE

AD = DE(given)

angle of ADE

= 90 + 60

= 150

angle DAE = 15(base angle at isos triangle)

angle BAE = 75

= angle ABE

so,△ABE is an isosceles triangle both in case 1 and 2.



2).
A regular n-sided polygin has 5 more sides than a regular m-sided polygon,

and the sum of interior angles of the former is twice that of the latter. Fing the

values of m and n.


n = m + 5-----[1]

(n - 2) x 180 = 2 x (m - 2) x 180-------[2]

sub [1] into [2]

(m + 5 - 2) = 2(m - 2)

m + 3 = 2m - 4

m = 7

n = 12