Geometry problem answers6 (50..59)

50.


calculate the ratio

Solution



51.
Below is pictured a square and a quarter arc.
Calculate the ratio of the blue and red areas.



Solution



52.
Below is pictured a slightly overturned rectangle.
Prove that the colored areas are equal.



Solution





53.
Calculate the area of square ABCD.



Solution



54.
AB=4.
Calculate the area of the colored triangle.



Solution

Triangles ABE and ADF are congruent (case edge,angle,angle) so BE=FD , AB=AD.
ABCD is a square.
Let BE=x.
Apply Pythagoras lemma:


55.
In picture below the medians are perpendicular.
Calculate side x.



Solution

The medians intersect at ratio 2:1 counted from the angle.
Let DF=a, EF=b.
AD=22, EC=19.
Apply Pythagoras lemma:


56.
Calculate angle x in the circle arc.



Solution

Observe isosceles traingles.



Sequence of calculations:
    LMCB = LMBC = 71
    LMCD = 140 - 71 = 69
    LMDC = 69
    LBMC = 180 - 2.71 = 38
    LCMD = 180 - 2. 69 - 42
    LAME = 180 - 2.65 = 50
    LDME = 180 - 50 - 42 - 38 = 50
    LMDE = (180 - 50)/2 = 65
    x = 69 + 65 = 134
57.


Solution



58.


The colored rectangles are identical.
a.
Calculated angle α

The area of a colored rectangle is 10.
b.
Calculate the area of rectangle ABCD.

Solution

a.



b.
[FGS]=[AES]
[ACD]=[FGC]+[AEFD]=10+5=15
[ABCD]=2.15=30
59.
Given is rectangle ABCD and a circle that contacts sides AB and AD.
C is on this circle.



Question: find f(x)= area of ABCD.

Solution