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UPSC CDS Math Area and Perimeter Questions
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© examsnet.com
Question : 21
Total: 116
The angles of a triangle are in the ratio 1 : 1 : 4 . If the perimeter of the triangle is k times its largest side, then what is the value of k ?
[2019 CDS-I]
1 +
2
√
3
1 -
2
√
3
2 +
2
√
3
2
Validate
Solution:
Let the sides of triangle opposite to ∠A, ∠B and ∠C be a, b and c, respectively.
Let ∠A, ∠B an and ∠C be 4x,x and x respectively.
4
x
+
x
+
x
=
180
°
⇒
x
=
30
°
Since ∠B = ∠C,therefore, b = c.
Now,
a
+
2
b
=
k
×
a
⇒
k
=
1
+
2
(
b
a
)
=
1
+
2
(
sin
x
sin
4
x
)
=
1
+
2
(
sin
30
∘
sin
120
∘
)
sin
30
∘
=
1
2
sin
120
∘
=
sin
60
∘
=
√
3
2
∴
k
=
1
+
2
×
1
2
×
2
√
3
=
1
+
2
√
3
© examsnet.com
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