NDA 2026 Mathematics PYQ — The angles , and of a triangle are in the ratio . If the longest … | Mathem Solvex | Mathem Solvex
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NDA 2026 — Mathematics PYQ
NDA | Mathematics | 2026
The angles A,B, and C of a triangle are in the ratio 1:1:4. If the longest side of the triangle is 3 units, then what is the perimeter of the triangle?
एक त्रिभुज के कोण A, B और C, 11:4 के अनुपात में हैं। यदि त्रिभुज की सबसे लंबी भुजा की लंबाई 3 इकाई है, तो त्रिभुज का परिमाप क्या है ?
Choose the correct answer:
A.
3+3 units
B.
3+23 units
(Correct Answer)
C.
3+33 units
D.
6+3 units
Correct Answer:
3+23 units
Explanation
Step 1: Determine the angles of the triangle
Let the angles be x,x, and 4x. Since the sum of angles in a triangle is 180∘:
x+x+4x=180∘
6x=180∘⟹x=30∘
Thus, the angles are A=30∘,B=30∘, and C=120∘.
Step 2: Identify the sides using the Sine Rule
The longest side is opposite the largest angle. Since 120∘ is the largest angle, the side c=3. According to the Sine Rule:
sinAa=sinBb=sinCc
sin30∘a=sin30∘b=sin120∘3
Since sin30∘=21 and sin120∘=sin(180∘−60∘)=sin60∘=23:
1/2a=1/2b=3/23
a=b=3/23⋅(1/2)=33=3
Step 3: Calculate the perimeter
The perimeter P is the sum of all sides:
P=a+b+c
P=3+3+3=3+23
Final Answer:
The perimeter of the triangle is 3+23 units. Therefore, the correct option is (b).
Explanation
Step 1: Determine the angles of the triangle
Let the angles be x,x, and 4x. Since the sum of angles in a triangle is 180∘:
x+x+4x=180∘
6x=180∘⟹x=30∘
Thus, the angles are A=30∘,B=30∘, and C=120∘.
Step 2: Identify the sides using the Sine Rule
The longest side is opposite the largest angle. Since 120∘ is the largest angle, the side c=3. According to the Sine Rule:
sinAa=sinBb=sinCc
sin30∘a=sin30∘b=sin120∘3
Since sin30∘=21 and sin120∘=sin(180∘−60∘)=sin60∘=23:
1/2a=1/2b=3/23
a=b=3/23⋅(1/2)=33=3
Step 3: Calculate the perimeter
The perimeter P is the sum of all sides:
P=a+b+c
P=3+3+3=3+23
Final Answer:
The perimeter of the triangle is 3+23 units. Therefore, the correct option is (b).
