MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - Middle School Tournaments
LMT
2023 LMT Fall
3A
3A
Part of
2023 LMT Fall
Problems
(1)
2023 Fall Theme p3A
Source:
12/23/2023
A rectangular tea bag
P
A
R
T
PART
P
A
RT
has a logo in its interior at the point
Y
Y
Y
. The distances from
Y
Y
Y
to
P
T
PT
PT
and
P
A
PA
P
A
are
12
12
12
and
9
9
9
respectively, and triangles
△
P
Y
T
\triangle PYT
△
P
Y
T
and
△
A
Y
R
\triangle AYR
△
A
Y
R
have areas
84
84
84
and
42
42
42
respectively. Find the perimeter of pentagon
P
A
R
T
Y
PARTY
P
A
RT
Y
.Proposed by Muztaba Syed Solution.
78
\boxed{78}
78
Using the area and the height in
△
P
Y
T
\triangle PYT
△
P
Y
T
, we see that
P
T
=
14
PT = 14
PT
=
14
, and thus
A
R
=
14
AR = 14
A
R
=
14
, meaning the height from
Y
Y
Y
to
A
R
AR
A
R
is
6
6
6
. This means
P
A
=
T
R
=
18
PA = TR = 18
P
A
=
TR
=
18
. By the Pythagorean Theorem
P
Y
=
1
2
2
+
9
2
=
15
PY=\sqrt{12^2+9^2} = 15
P
Y
=
1
2
2
+
9
2
=
15
and
Y
T
=
1
2
2
+
5
2
=
13
YT =\sqrt{12^2 +5^2} = 13
Y
T
=
1
2
2
+
5
2
=
13
. Combining all of these gives us an answer of
18
+
14
+
18
+
13
+
15
=
78
18+14+18+13+15 = \boxed{78}
18
+
14
+
18
+
13
+
15
=
78
.
2023
theme
FAlL
geo