Difference between revisions of "1998 JBMO Problems/Problem 2"
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Construct <math>AD</math> and <math>AC</math> to partition the figure into <math>ABC</math>, <math>ACD</math> and <math>ADE</math>. | Construct <math>AD</math> and <math>AC</math> to partition the figure into <math>ABC</math>, <math>ACD</math> and <math>ADE</math>. | ||
− | Rotate <math>ADE</math> with centre <math>A</math> such that <math>AE</math> coincides with <math>AB</math> and <math>AD</math> is mapped to <math>AD'</math>. Hence the area of the pentagon is still preserved and it suffices to find the area of the | + | Rotate <math>ADE</math> with centre <math>A</math> such that <math>AE</math> coincides with <math>AB</math> and <math>AD</math> is mapped to <math>AD'</math>. Hence the area of the pentagon is still preserved and it suffices to find the area of the quadrilateral <math>AD'CD</math>. |
Hence <math>[AD'C]</math> = <math>\frac{1}{2}</math> (<math>D'E + BC</math>)<math>AB</math>= <math>\frac{1}{2}</math> | Hence <math>[AD'C]</math> = <math>\frac{1}{2}</math> (<math>D'E + BC</math>)<math>AB</math>= <math>\frac{1}{2}</math> |
Latest revision as of 08:31, 2 July 2020
Problem 2
Let be a convex pentagon such that
,
and
. Compute the area of the pentagon.
Solutions
Solution 1
Let
Let
Applying cosine rule to we get:
Substituting we get:
From above,
Thus,
So, area of =
Let be the altitude of
from
.
So
This implies .
Since is a cyclic quadrilateral with
,
is congruent to
.
Similarly
is a cyclic quadrilateral and
is congruent to
.
So area of + area of
= area of
.
Thus area of pentagon
= area of
+ area of
+ area of
=
By
Solution 2
Let . Denote the area of
by
.
can be found by Heron's formula.
Let .
Total area .
By durianice
Solution 3
Construct and
to partition the figure into
,
and
.
Rotate with centre
such that
coincides with
and
is mapped to
. Hence the area of the pentagon is still preserved and it suffices to find the area of the quadrilateral
.
Hence =
(
)
=
Since =
,
=
and
=
, by SSS Congruence,
and
are congruent, so
=
So the area of pentagon .
- SomebodyYouUsedToKnow
See Also
1998 JBMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |