An isosceles triangle has a vertex angle of 30 degrees, and each of the two equal sides measures 15 units. Find the length of the base of the triangle using trigonometric functions. Then, calculate the area of the triangle. Finally, suppose a circumcircle passes through all three vertices of the triangle. Find the radius of this circumcircle.

To solve this problem, let's break it down into parts and address each requirement step by step.

### 1. Finding the Length of the Base
Given:
- Vertex angle \( \theta = 30^\circ \)
- Equal sides \( a = 15 \) units

Since the triangle is isosceles, the vertex angle \( \theta \) divides the triangle into two congruent right triangles. The angle at each of the two base angles is \( 15^\circ \). We will use the trigonometric function \( \sin \) to find half of the base length.

#### Step-by-Step Calculation:
- The height \( h \) of the triangle can be found using \( \cos(15^\circ) \):
  \[
  h = a \cdot \cos(15^\circ) = 15 \cdot \cos(15^\circ)
  \]
- Half of the base length \( \frac{b}{2} \) can be found using \( \sin(15^\circ) \):
  \[
  \frac{b}{2} = a \cdot \sin(15^\circ) = 15 \cdot \sin(15^\circ)
  \]
Thus, the total base \( b \) is:
\[
b = 2 \cdot (15 \cdot \sin(15^\circ))
\]

**Calculating:**
\[
\sin(15^\circ) \approx 0.2588
\]
\[
b = 2 \cdot 15 \cdot 0.2588 \approx 7.764 \text{ units}
\]

### 2. Calculating the Area of the Triangle
The area \( A \) of the triangle can be calculated using:
\[
A = \frac{1}{2} \cdot \text{base} \cdot \text{height}
\]
We already found that the height \( h \) is \( 15 \cdot \cos(15^\circ) \), and the base \( b = 7.764 \) units.

\[
\cos(15^\circ) \approx 0.9659
\]
\[
h = 15 \cdot 0.9659 \approx 14.489 \text{ units}
\]
Thus:
\[
A = \frac{1}{2} \cdot 7.764 \cdot 14.489 \approx 56.27 \text{ square units}
\]

### 3. Finding the Radius of the Circumcircle
For a triangle, the circumradius \( R \) can be found using:
\[
R = \frac{abc}{4A}
\]
where \( a = b = 15 \) and \( c = 7.764 \).

#### Step-by-Step Calculation:
\[
R = \frac{15 \cdot 15 \cdot 7.764}{4 \cdot 56.27}
\]
\[
R = \frac{1749.45}{225.08} \approx 7.77 \text{ units}
\]

### Summary of Results:
1. **Length of the base:** \( \approx 7.764 \) units
2. **Area of the triangle:** \( \approx 56.27 \) square units
3. **Radius of the circumcircle:** \( \approx 7.77 \) units

Would you like more details on any of these calculations, or do you have further questions?

### Related Questions:
1. How can we find the area using a different trigonometric approach?
2. What would happen if the vertex angle were doubled? How would that affect the triangle's properties?
3. How can the radius of the circumcircle be verified using other formulas?
4. What is the formula for the height of an isosceles triangle, and how is it derived?
5. Can the circumradius formula be generalized for any triangle?

### Tip:
When dealing with isosceles triangles, splitting them into two right triangles simplifies the calculations, especially when using trigonometric functions.

Learn how to solve for the base length, area, and circumradius of an isosceles triangle with a vertex angle of 30 degrees and equal sides of 15 units. This problem covers key trigonometric functions and geometric principles, making it ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation