Difference between revisions of "2023 AIME I Problems/Problem 8"
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==Problem== | ==Problem== | ||
− | Rhombus <math>ABCD</math> has <math>\angle BAD<90^\circ</math> | + | Rhombus <math>ABCD</math> has <math>\angle BAD < 90^\circ.</math> There is a point <math>P</math> on the incircle of the rhombus such that the distances from <math>P</math> to the lines <math>DA,</math> <math>AB,</math> and <math>BC</math> are <math>9,</math> <math>5,</math> and <math>16,</math> respectively. Find the perimeter of <math>ABCD.</math> |
==Solution 1== | ==Solution 1== |
Revision as of 02:57, 9 February 2023
Problem
Rhombus has
There is a point
on the incircle of the rhombus such that the distances from
to the lines
and
are
and
respectively. Find the perimeter of
Solution 1
Denote by the center of
.
We drop an altitude from
to
that meets
at point
.
We drop altitudes from
to
and
that meet
and
at
and
, respectively.
We denote
.
We denote the side length of
as
.
Because the distances from to
and
are 16 and 9, respectively, and
, the distance between each pair of two parallel sides of
is
.
Thus,
and
.
We have
Thus, .
In , we have
.
Thus,
Taking the imaginary part of this equation and plugging and
into this equation, we get
We have
Because is on the incircle of
,
. Plugging this into (1), we get the following equation
By solving this equation, we get and
.
Therefore,
.
Therefore, the perimeter of is
.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2023 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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