MathDB

2023 Fall Theme p3A

Source:

December 23, 2023

2023themeFAlLgeo

Problem Statement

A rectangular tea bag

PART

has a logo in its interior at the point

Y

. The distances from

Y

to

PT

and

PA

are

12

and

9

respectively, and triangles

\triangle PYT

and

\triangle AYR

have areas

84

and

42

respectively. Find the perimeter of pentagon

PARTY

.
Proposed by Muztaba Syed
Solution.

\boxed{78}

Using the area and the height in

\triangle PYT

, we see that

PT = 14

, and thus

AR = 14

, meaning the height from

Y

to

AR

is

6

. This means

PA = TR = 18

. By the Pythagorean Theorem

PY=\sqrt{12^2+9^2} = 15

and

YT =\sqrt{12^2 +5^2} = 13

. Combining all of these gives us an answer of

18+14+18+13+15 = \boxed{78}

.

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