https://file.aiquickdraw.com/mathbot/file/1729920028984z5x4v37y.jpg

To solve this problem, let's go through the given information step-by-step:

1. **Circumradius (R)** of triangle \( ABC \) is given as \( R = 5 \) cm.
2. **Right Angle at \( \angle ADB \)**: Since \( \angle ADB \) is a right angle, \( D \) lies on the circumcircle and \( AD \) acts as the angle bisector of \( \angle A \).
3. **Ratio of the Sides \( AC:CB = 3:4 \)**: We can denote \( AC = 3x \) and \( CB = 4x \) for some constant \( x \).

Since \( \triangle ABC \) is inscribed in a circle with \( \angle ADB = 90^\circ \), this implies that \( \triangle ABC \) is a **right triangle** with \( \angle ACB = 90^\circ \) by the property of the angle in a semicircle (Thales' theorem).

### Step 1: Determine the Side Lengths Using the Circumradius

For a right triangle, the circumradius \( R \) is half the hypotenuse \( AB \):
\[
R = \frac{AB}{2} \Rightarrow AB = 2 \cdot R = 2 \cdot 5 = 10 \text{ cm}.
\]

### Step 2: Set up a Relationship Using the Side Ratios

From the ratio \( AC : CB = 3 : 4 \), let:
\[
AC = 3x \quad \text{and} \quad CB = 4x.
\]

Since \( \triangle ABC \) is a right triangle, we can use the Pythagorean theorem:
\[
AB^2 = AC^2 + CB^2.
\]
Substituting \( AB = 10 \), \( AC = 3x \), and \( CB = 4x \), we get:
\[
10^2 = (3x)^2 + (4x)^2.
\]
\[
100 = 9x^2 + 16x^2.
\]
\[
100 = 25x^2.
\]
\[
x^2 = 4 \Rightarrow x = 2.
\]

### Step 3: Calculate \( AC \) and \( CB \)

Now that we have \( x = 2 \):
\[
AC = 3 \cdot 2 = 6 \text{ cm}, \quad CB = 4 \cdot 2 = 8 \text{ cm}.
\]

### Step 4: Calculate the Area of \( \triangle ABC \)

For a right triangle, the area is given by:
\[
\text{Area} = \frac{1}{2} \times AC \times CB.
\]
Substitute \( AC = 6 \) and \( CB = 8 \):
\[
\text{Area} = \frac{1}{2} \times 6 \times 8 = \frac{1}{2} \times 48 = 24 \text{ cm}^2.
\]

### Final Answer
The area of \( \triangle ABC \) is:
\[
\boxed{24 \text{ cm}^2}.
\]

Would you like more details on any specific step?

---

### Related Questions

1. How does the circumradius relate to the hypotenuse in a right triangle?
2. Can this method be applied if the triangle is not a right triangle?
3. How does the angle bisector property affect the solution here?
4. What would be the area if the ratio of \( AC \) to \( CB \) was different?
5. How can you verify the area using trigonometric methods?

---

### Tip
In any right triangle inscribed in a circle, the hypotenuse is always the diameter of the circle. This is a consequence of the Thales' theorem.

In the figure, the circumradius of triangle ABC is 5 cm. Let D be the intersection of the angle-bisector of angle A with the circumcircle of triangle ABC. It’s known that angle ADB is a right angle. In the triangle, the ratio of the sides AC : CB = 3 : 4. Then find the area of the triangle ABC.

Calculate the area of triangle ABC given its circumradius of 5 cm, a right angle at ADB, and a side ratio of AC:CB = 3:4. This problem uses key geometry concepts such as Thales' theorem and the Pythagorean theorem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation