A triangle with sides in the ratio 20 : 21 : 29 is inscribed in a circle with radius 5. The area (in sq. units) of the triangle is closest to

Given a triangle with sides in the ratio \( 20 : 21 : 29 \) inscribed in a circle with a radius of 5 units, we are tasked with finding the area of the triangle.

### Step 1: Identifying the sides
Since the sides are in the ratio \( 20 : 21 : 29 \), we can express the sides as:
- \( a = 20k \)
- \( b = 21k \)
- \( c = 29k \)

The side \( c = 29k \) is the longest, and it corresponds to the diameter of the circle because the triangle is inscribed in a circle. The relationship between the diameter and the circumradius \( R \) is given by:
\[
c = 2R
\]
Since the radius \( R = 5 \), we have:
\[
c = 2 \times 5 = 10
\]
Thus, \( 29k = 10 \), which gives \( k = \frac{10}{29} \).

### Step 2: Calculating the sides
Now, substitute \( k = \frac{10}{29} \) into the expressions for the sides:
\[
a = 20k = 20 \times \frac{10}{29} = \frac{200}{29}
\]
\[
b = 21k = 21 \times \frac{10}{29} = \frac{210}{29}
\]
\[
c = 29k = 10
\]

### Step 3: Using the area formula for a triangle inscribed in a circle
The formula for the area of a triangle inscribed in a circle is:
\[
\text{Area} = \frac{abc}{4R}
\]
Substituting the values of \( a \), \( b \), \( c \), and \( R \):
\[
\text{Area} = \frac{\frac{200}{29} \times \frac{210}{29} \times 10}{4 \times 5}
\]
Simplifying:
\[
\text{Area} = \frac{\frac{200 \times 210 \times 10}{29^2}}{20}
\]
\[
\text{Area} = \frac{200 \times 210 \times 10}{20 \times 29^2}
\]
\[
\text{Area} = \frac{420000}{29^2 \times 20} = \frac{420000}{16820} \approx 24.97 \, \text{sq. units}
\]

### Final Answer:
The area of the triangle is approximately \( 25 \, \text{sq. units} \).

Would you like more details on any of these steps?

Here are five related questions you might be interested in:
1. How do you find the circumradius of a triangle using other properties?
2. What is the significance of the inscribed circle in a triangle?
3. How can Heron's formula be applied to find the area of a triangle?
4. What are other methods for calculating the area of a triangle inscribed in a circle?
5. Can this method be extended to other polygons?

**Tip:** When a triangle is inscribed in a circle, and one side equals the diameter, the triangle is always a right triangle.

Discover how to find the area of a triangle with sides in the ratio 20:21:29 inscribed in a circle with radius 5. This problem involves geometric concepts like circumradius, side ratios, and the area formula for a triangle. Suitable for students in Grades 10-12.

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