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UPSC CDS Math Triangles Questions
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© examsnet.com
Question : 54
Total: 68
The angles of a triangle are in the ratio 4:1:1. Then the ratio of the largest side to the perimeter is
[2015 CDS-I]
2
3
1
2
+
√
3
√
3
2
+
√
3
2
1
+
√
3
Validate
Solution:
Let the angles be 4k, k and k, respectively.
Now,
4
k
+
k
+
k
=
180
°
⇒
6
k
=
180
°
⇒
k
=
30
°
Therefore, the angles of the triangle are 120°, 30° and 30°.
Let the sides opposite to theseangles be a, b and c. So, a is the largest side of the triangle.
∴
a
sin
120
°
=
b
sin
30
°
=
c
sin
30
°
⇒
a
(
√
3
2
)
=
b
(
1
2
)
=
c
(
1
2
)
(
sin
120
°
=
sin
60
°
=
√
3
2
)
⇒
a
√
3
=
b
1
=
c
1
=
a
+
b
+
c
√
3
+
1
+
1
⇒
a
a
+
b
+
c
=
√
3
2
+
√
3
© examsnet.com
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