Question

The area of an isosceles triangle is $9c{{m}^{2}}$. If the equal sides are $6cm$ in length, the angle between them is.
A. ${{60}^{0}}$
B. ${{30}^{0}}$
C. ${{90}^{0}}$
D. ${{45}^{0}}$

Answer 1

Hint: To solve this question, we will use the formula of the area of the triangle when two sides and an angle between are given. We will take any two sides as equal having value $6cm$ and any one angle which is included between them. We will substitute the length of the sides and area of the triangle in the formula and simplify it and determine the value of the angle.
Formula used:
Area of the triangle:
\[Area=\frac{1}{2}bc\sin A\]
Complete step-by-step solution:
We are given area of an isosceles triangle as $9 c{{m}^{2}}$ and the value of the length of the equal sides of the triangle as $6cm$ and we have to determine the angle between the equal sides.
Let us take the equal sides as $b,c$ that is $b=c=6 cm$.
Now we will use the area of the triangle and substitute the given value of area and sides of the triangle,
\[\begin{align}
  & Area=\frac{1}{2}bc\sin A \\
 & 9=\frac{1}{2}\times 6\times 6\sin A \\
 & 9=18\sin A \\
 & \frac{1}{2}=\sin A
\end{align}\]
We know that \[\sin {{30}^{0}}=\frac{1}{2}\] so,
\[\begin{align}
  & \sin {{30}^{0}}=\sin A \\
 & A={{30}^{0}}
\end{align}\]
The angle between the equal sides of the triangle is \[{{30}^{0}}\] when the area is $9c{{m}^{2}}$ and the equal sides are $6cm$ in length. Hence the correct option is (B).
Note:

A triangle having two equal sides and two equal angles is an isosceles triangle. If we draw a perpendicular from the apex of an isosceles triangle which is also called as the line of Symmetry, then it will divide the triangle into two congruent triangles which means both the triangle will be equal in length and angles.
According to the isosceles property, the angles opposite to sides which are equal in length are also equal.