22.
A semicircle of radius
r
is drawn with centre
V
and
diameter
UW.
The line
UW
is then extended to the
point
X,
such that the
UW
and
WX
are of equal length.
An arc of the circle with centre
X
and radius 4
r
is then
drawn so that the line
XY
is a tangent to the semicircle
at
Z,
as shown. What, in terms of
r,
is the area of triangle
YVW?
A
9
4
2
r
B
3
2
2
r
C
2
r
D
3
4
2
r
E
2
2
r
Solution:
B
Since
WX
UW
=
, we have
.
2
r
UW
WX
=
=
As
V
is
the centre of the circle with
UW
as diameter,
.
r
VZ
VW
=
=
Hence
r
WX
VW
VX
3
=
+
=
. The triangles
YVX
and
VVW
have the same height and
VX
VW
3
1
=
. Therefore
)
(
area
)
(
area
3
1
YVX
YVW
∆
=
∆
.
As
VZ
is a radius of the
semicircle, and
XY
is a tangent to the semicircle at
Z
, the lines
VZ
and
XY
are perpendicular. Therefore
.
2
)
4
.
(
.
)
(
area
2
2
1
2
1
r
r
r
XY
VZ
YVX
=
=
=
∆
Hence,
.
)
2
(
)
(
area
)
(
area
2
3
2
2
3
1
3
1
r
r
YVX
YVW
=
=
∆
=
∆
U
V
W
X
Y
Z
U
V
W
X

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