MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canadian Open Math Challenge
2017 Canadian Open Math Challenge
C3
C3
Part of
2017 Canadian Open Math Challenge
Problems
(1)
2017 COMC C3
Source:
10/12/2018
Source: 2017 Canadian Open Math Challenge, Problem C3 —-- Let
X
Y
Z
XYZ
X
Y
Z
be an acute-angled triangle. Let
s
s
s
be the side-length of the square which has two adjacent vertices on side
Y
Z
YZ
Y
Z
, one vertex on side
X
Y
XY
X
Y
and one vertex on side
X
Z
XZ
XZ
. Let
h
h
h
be the distance from
X
X
X
to the side
Y
Z
YZ
Y
Z
and let
b
b
b
be the distance from
Y
Y
Y
to
Z
Z
Z
.[asy] pair S, D; D = 1.27; S = 2.55; draw((2, 4)--(0, 0)--(7, 0)--cycle); draw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle); label("
X
X
X
",(2,4),N); label("
Y
Y
Y
",(0,0),W); label("
Z
Z
Z
",(7,0),E); [/asy](a) If the vertices have coordinates
X
=
(
2
,
4
)
X = (2, 4)
X
=
(
2
,
4
)
,
Y
=
(
0
,
0
)
Y = (0, 0)
Y
=
(
0
,
0
)
and
Z
=
(
4
,
0
)
Z = (4, 0)
Z
=
(
4
,
0
)
, find
b
b
b
,
h
h
h
and
s
s
s
. (b) Given the height
h
=
3
h = 3
h
=
3
and
s
=
2
s = 2
s
=
2
, find the base
b
b
b
. (c) If the area of the square is
2017
2017
2017
, determine the minimum area of triangle
X
Y
Z
XYZ
X
Y
Z
.
Comc
2017 COMC