Analyzing the Triangle Geometry
The question asks for the area of a triangle with a given base and two equal side lengths. This describes an isosceles triangle.
- Base ($b$): 10 cm
- Equal Sides ($s$): 6 cm
The area of a triangle is calculated using the formula:
$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $
We need to find the height ($h$) of the triangle.
Calculating the Triangle Height
In an isosceles triangle, the altitude drawn to the base bisects the base.
- The base is divided into two segments, each of length $\frac{10}{2} = 5$ cm.
- The altitude forms a right-angled triangle with one of the equal sides (hypotenuse = 6 cm) and half the base (one leg = 5 cm). The altitude is the other leg.
Using the Pythagorean theorem ($a^2 + b^2 = c^2$):
$ 5^2 + h^2 = 6^2 $
$ 25 + h^2 = 36 $
$ h^2 = 36 - 25 $
$ h^2 = 11 $
$ h = \sqrt{11} \text{ cm} $
Calculating the Triangle Area
Now, substitute the base and the calculated height into the area formula:
$ \text{Area} = \frac{1}{2} \times b \times h $
$ \text{Area} = \frac{1}{2} \times 10 \text{ cm} \times \sqrt{11} \text{ cm} $
$ \text{Area} = 5 \times \sqrt{11} \text{ sq. cm} $
The area of the triangle is $5\sqrt{11}$ sq. cm.
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 triangle $ABC$, the length of the side $AB$ is 2m, the angle at $C$ is $60^\circ$ and the angle at $B$ is $30^\circ$, what is the length of side $AC$?
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 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 triangle $ABC$, the length of the side $AB$ is 2m, the angle at $C$ is $60^\circ$ and the angle at $B$ is $30^\circ$, what is the length of side $AC$?
