isosceles triangle has base 6cm and tan of its base angle equal square root of 5.whats the area of the triangle

To find the area of the isosceles triangle, we can follow these steps:

### Given:
- Base of the isosceles triangle \( b = 6 \, \text{cm} \)
- Tangent of the base angle \( \tan(\theta) = \sqrt{5} \)

### Step 1: Identify the height of the triangle.
Since it's an isosceles triangle, the height (\(h\)) will split the base into two equal halves, each of 3 cm.

We can use the relationship for the tangent of the base angle (\( \theta \)):

\[
\tan(\theta) = \frac{\text{opposite (height)}}{\text{adjacent (half of the base)}} = \frac{h}{3}
\]

Given \( \tan(\theta) = \sqrt{5} \), we have:

\[
\sqrt{5} = \frac{h}{3}
\]

Solving for \( h \):

\[
h = 3 \times \sqrt{5} = 3\sqrt{5} \, \text{cm}
\]

### Step 2: Calculate the area of the triangle.
The area \( A \) of a triangle is given by the formula:

\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

Substituting the values:

\[
A = \frac{1}{2} \times 6 \times 3\sqrt{5}
\]

\[
A = 9\sqrt{5} \, \text{cm}^2
\]

Thus, the area of the isosceles triangle is \( 9\sqrt{5} \, \text{cm}^2 \), or approximately \( 20.12 \, \text{cm}^2 \).

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Would you like a more detailed breakdown or have any questions?

Here are some related questions for deeper understanding:
1. How do we find the height of a general isosceles triangle given the base and angles?
2. What is the relationship between the tangent of an angle and the sides of a right triangle?
3. Can you find the area of an isosceles triangle given only its sides?
4. How would the area change if the base angle were larger or smaller?
5. How does the trigonometric function \( \tan(\theta) \) behave for different angles?

**Tip:** In an isosceles triangle, the height always divides the base into two equal parts, which helps simplify the use of trigonometric ratios.

Isosceles triangle has base 6 cm and tan of its base angle equal to square root of 5. What's the area of the triangle?

Find the area of an isosceles triangle with a base of 6 cm and a base angle where the tangent is the square root of 5. This problem involves trigonometric concepts and geometry suitable for students in Grades 8-10, with a step-by-step solution provided.

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