Which Triangle is Larger?
is a regular pentagon of side length 1. is an equilateral triangle with a side congruent to the pentagon's diagonal. Which triangle, red or blue, has the larger area? For extra credit, provide closed-form representations of the areas of the red and blue triangles.
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1 solution
All angles are in degrees.
Line segment
E
H
divides
△
A
B
E
into two congruent right triangles. This gives the length of
A
B
as
A
B
=
2
⋅
1
⋅
cos
(
3
6
)
=
2
cos
(
3
6
)
Using the sine rule on
△
B
F
G
, we can calculate the sides
F
G
and
B
G
.
sin
(
6
0
)
1
=
sin
(
1
2
)
F
G
=
sin
(
1
0
8
)
B
G
F
G
=
sin
(
6
0
)
sin
(
1
2
)
B
G
=
sin
(
6
0
)
sin
(
1
0
8
)
Since
△
A
B
C
is equilateral,
B
G
+
G
C
=
A
B
. Consequently,
G
C
=
A
B
−
B
G
=
2
cos
(
3
6
)
−
sin
(
6
0
)
sin
(
1
0
8
)
Use the area formula
A
=
2
1
a
b
sin
(
C
)
on both the blue and red triangles.
A
(
blue
)
=
2
1
⋅
1
⋅
sin
(
6
0
)
sin
(
1
2
)
⋅
sin
(
1
0
8
)
≈
0
.
1
1
4
A
(
red
)
=
2
1
⋅
(
2
cos
(
3
6
)
−
sin
(
6
0
)
sin
(
1
0
8
)
)
2
⋅
sin
(
6
0
)
≈
0
.
1
1
7
Therefore, the red triangle has a larger area than the blue triangle.