
In triangle $ABC$ and $DEF$, $AB=DE,~AC=EF$ and $\angle A=2\angle E$. Two triangles will have the same area, if angle $A$ is equal to.
A. $\dfrac{\pi }{3}$
B. $\dfrac{\pi }{2}$
C. $\dfrac{2\pi }{3}$
D. $\dfrac{5\pi }{6}$
Answer
162.6k+ views
Hint: To solve this question, we will use the given condition of areas being equal of both the triangles. We will use the formula of areas for both the triangles and equate them with each other. We will then simplify the equation using trigonometric formulas and get an equation in terms of the angle. We will then use $\angle A=2\angle E$ and find the value of angle $E$. After this we will find the value of angle $A$ using the substitution method.
Formula Used: Area of the triangle with two sides and angle is calculated with formula $Area=\dfrac{1}{2}bc\sin A$, where $b,c$ are the sides and $A$ is angle.
$\sin 2A=2\sin A\cos A$
Complete step by step solution: We are given a triangle $ABC$ and $DEF$ in which $AB=DE,~AC=EF$ and $\angle A=2\angle E$. We have to find the value of the angle $A$, if both the triangles have the same area.
We will draw both the triangles $ABC$ and $DEF$.
Now as given both the areas of the triangles are the same. So we will use the formula of calculating area with two sides and an angle.
$\begin{align}
& \Delta ABC=\Delta DEF \\
& \dfrac{1}{2}\times AB\times AC\times \sin A=\dfrac{1}{2}\times DE\times EF\times \sin E
\end{align}$
As$AB=DE,AC=EF$, the equation will be,
$\sin A=\sin E$
We will now substitute the value of angle $\angle A=2\angle E$ and evaluate using the formula of double angle of sine.
$\begin{align}
& \sin 2E=\sin E \\
& 2\sin E\cos E=\sin E \\
& \cos E=\dfrac{1}{2} \\
& \cos E=\cos \dfrac{\pi }{3} \\
& E=\dfrac{\pi }{3}
\end{align}$
To calculate the value of angle $A$ we will substitute the value of angle $E$ that is $E=\dfrac{\pi }{3}$ in $\angle A=2\angle E$.
$\begin{align}
& \angle A=2\angle E \\
& \angle A=\dfrac{2\pi }{3} \\
\end{align}$
The value of angle $A$ of triangle $ABC$ is $\angle A=\dfrac{2\pi }{3}$ when area of the triangles $ABC$ and $DEF$ are same, $AB=DE,~AC=EF$ and $\angle A=2\angle E$. Hence the correct option is (C).
Note: In solving these questions, we must be aware of the formula of the area of the triangle when two sides and an angle is given as well as also have required knowledge of the trigonometric table values and formulas.
Formula Used: Area of the triangle with two sides and angle is calculated with formula $Area=\dfrac{1}{2}bc\sin A$, where $b,c$ are the sides and $A$ is angle.
$\sin 2A=2\sin A\cos A$
Complete step by step solution: We are given a triangle $ABC$ and $DEF$ in which $AB=DE,~AC=EF$ and $\angle A=2\angle E$. We have to find the value of the angle $A$, if both the triangles have the same area.
We will draw both the triangles $ABC$ and $DEF$.
Now as given both the areas of the triangles are the same. So we will use the formula of calculating area with two sides and an angle.
$\begin{align}
& \Delta ABC=\Delta DEF \\
& \dfrac{1}{2}\times AB\times AC\times \sin A=\dfrac{1}{2}\times DE\times EF\times \sin E
\end{align}$
As$AB=DE,AC=EF$, the equation will be,
$\sin A=\sin E$
We will now substitute the value of angle $\angle A=2\angle E$ and evaluate using the formula of double angle of sine.
$\begin{align}
& \sin 2E=\sin E \\
& 2\sin E\cos E=\sin E \\
& \cos E=\dfrac{1}{2} \\
& \cos E=\cos \dfrac{\pi }{3} \\
& E=\dfrac{\pi }{3}
\end{align}$
To calculate the value of angle $A$ we will substitute the value of angle $E$ that is $E=\dfrac{\pi }{3}$ in $\angle A=2\angle E$.
$\begin{align}
& \angle A=2\angle E \\
& \angle A=\dfrac{2\pi }{3} \\
\end{align}$
The value of angle $A$ of triangle $ABC$ is $\angle A=\dfrac{2\pi }{3}$ when area of the triangles $ABC$ and $DEF$ are same, $AB=DE,~AC=EF$ and $\angle A=2\angle E$. Hence the correct option is (C).
Note: In solving these questions, we must be aware of the formula of the area of the triangle when two sides and an angle is given as well as also have required knowledge of the trigonometric table values and formulas.
Recently Updated Pages
How To Find Mean Deviation For Ungrouped Data

Difference Between Molecule and Compound: JEE Main 2024

Ammonium Hydroxide Formula - Chemical, Molecular Formula and Uses

Difference Between Area and Surface Area: JEE Main 2024

Difference Between Work and Power: JEE Main 2024

Difference Between Acetic Acid and Glacial Acetic Acid: JEE Main 2024

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More
