Question

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 1

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.