To find the length of side \(AC\) in triangle \(ABC\), we will employ the Law of Sines, which states:
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
From the question, we know:
- The length of side \(AB = c = 2 \text{ m}\)
- Angle at \(C = 60^\circ\)
- Angle at \(B = 30^\circ\)
First, we calculate the angle at \(A\) using the angle sum property of triangles:
\(A = 180^\circ - C - B = 180^\circ - 60^\circ - 30^\circ = 90^\circ\)
Now, apply the Law of Sines:
\(\frac{AC}{\sin B} = \frac{AB}{\sin C}\)
Substitute known values:
\(\frac{AC}{\sin 30^\circ} = \frac{2}{\sin 60^\circ}\)
We know:
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
Substituting these values gives:
\(\frac{AC}{\frac{1}{2}} = \frac{2}{\frac{\sqrt{3}}{2}}\)
Solving for \(AC\):
\(AC = \frac{2 \times \frac{1}{2}}{\frac{\sqrt{3}}{2}}\)
\(AC = \frac{2}{\sqrt{3}}\)
Thus, the length of side \(AC\) is \(\frac{2}{\sqrt{3}} \text{ m}\).
Therefore, the correct answer is \(\frac{2}{\sqrt{3}} \text{ m}\).
Similar Questions
In the given figure, if \(RS = 3\sqrt{3} \text{ cm}\) and \(RPS\) is an equilateral triangle, then find the value of \(QR\)
- In a right angled triangle, with angle at \(A\) being \(90^\circ\), the side \(AB\) is of length 3cm and \(BC\) is 12 cm. What is the length of side \(AC\)?
For a pair of similar triangles shown in the figure, the angles made at $B$ and $R$ are same in both the triangles $BAC$ and $RAQ$. If the ratio of the areas of the two triangles is 2:1, and if the area of $BAC$ is $100\text{ cm}^2$ then what is the area of $RAQ$ in sq.cm.?
In a right angled triangle, with angle at A being , the side $AB$ is of length 4cm and $BC$ is 15 cm. What is the length of side $AC$?
- Triangles $PQR$ and $ABC$ are similar triangles.
Triangle $ABC$ have the following sides.
$AB = 36\text{ cm}$
$AC = 20\text{ cm}$
$BC = 48\text{ cm}$
Triangle $PQR$ have the following sides:
$PQ = 9\text{ cm}$
$PR = 5\text{ cm}$
What is the length of side $RQ$ =? - What is the area of the triangle in sq. cm. with a base of 10 cm and sides of length 6 cm each?
What is the angle at $B$ in the triangle $ABC$, if the angle made at $K$ is $30^\circ$ and the triangles shown in the figure are similar?
- In a triangle the length of the sides, $AB$ is 3m and $BC$ is 5m what is the length of $AC$, if the angle formed at $B$ is $60^\circ$
- In a triangle \(\Delta ABC\ AB= 13\text{ cm}, BC=84\text{cm}, \text{and } \angle B=90^\circ\), then length of AC will be
- Which of the following is not a Pythagorean triplet?
Important Questions from Triangles
What is \(AB + BC\) equal to?
In the given figure, if \(RS = 3\sqrt{3} \text{ cm}\) and \(RPS\) is an equilateral triangle, then find the value of \(QR\)
- In a right angled triangle, with angle at \(A\) being \(90^\circ\), the side \(AB\) is of length 3cm and \(BC\) is 12 cm. What is the length of side \(AC\)?
For a pair of similar triangles shown in the figure, the angles made at $B$ and $R$ are same in both the triangles $BAC$ and $RAQ$. If the ratio of the areas of the two triangles is 2:1, and if the area of $BAC$ is $100\text{ cm}^2$ then what is the area of $RAQ$ in sq.cm.?
In a right angled triangle, with angle at A being , the side $AB$ is of length 4cm and $BC$ is 15 cm. What is the length of side $AC$?
